A Study to Compare Treatments for a Type of Kidney Cancer Called TFE/Translocation Renal Cell Carcinoma (tRCC)

Inclusion Criteria: * Patients must be \>= 12 months at enrollment * Patients must have a body surface area (BSA) \>= 0.53 m\^2 * Histologically confirmed unresectable or metastatic translocation morphology renal cell carcinoma diagnosed using World Health Organization (WHO)-defined criteria. Patients may be newly diagnosed or have received prior cancer therapy * Patients must have had histologic verification of the malignancy * Patients must have measurable disease, documented by clinical, radiographic, or histologic criteria as defined by Response Evaluation Criteria in Solid Tumors (RECIST) version (v)1.1 * Patients must have a tumor showing the appropriate morphologic appearance, and either confirmed TFE3 nuclear protein expression by immunohistochemistry with appropriate positive and negative controls performed at a Clinical Laboratory Improvement Act (CLIA)-certified laboratory, or evidence of TFE3 or TFEb translocation by either fluorescence in situ hybridization (FISH) or reverse transcriptase- polymerase chain reaction (RT-PCR) performed at a CLIA-certified laboratory. For TFE3 immunohistochemistry, any nuclear positivity in the presence of appropriate positive and negative controls should be considered as evidence of TFE3 immunohistochemical expression. NOTE: If the institution is unable to perform these studies, unstained slides may be submitted to Dr. Elizabeth Perlman, who will perform TFE3 analysis at no charge. The slide will be returned to the referring institution for local evaluation, to be included in their institutional report * Patients must have a performance status corresponding to Eastern Cooperative Oncology Group (ECOG) scores of 0, 1 or 2. Use Karnofsky for patients \> 16 years of age and Lansky for patients =\< 16 years of age * Patients must have a life expectancy of \>= 8 weeks * Patients must have fully recovered from the acute toxic effects of all prior chemotherapy, immunotherapy, or radiotherapy prior to entering this study * Myelosuppressive chemotherapy: Must not have received within 2 weeks of entry onto this study (6 weeks if prior nitrosourea) * Immunotherapy: Must not have received within 4 weeks of entry onto this study * Biologic (anti-neoplastic agent): At least 7 days since the completion of therapy with a biologic agent * Radiation therapy (RT): \>= 2 weeks for local palliative RT (small port); \>= 6 months must have elapsed if prior craniospinal RT or if \>= 50% radiation of pelvis; \>= 6 weeks must have elapsed if other substantial bone marrow (BM) radiation * Peripheral absolute neutrophil count (ANC) \>= 1000/uL (performed within 7 days prior to enrollment) * Platelet count \>= 75,000/uL (transfusion independent) (performed within 7 days prior to enrollment) * Hemoglobin \>= 8.0 g/dL (may receive red blood cell \[RBC\] transfusions) (performed within 7 days prior to enrollment) * Urine protein: =\< 30 mg/dL in urinalysis or =\< 1+ on dipstick, unless quantitative protein is \< 1000 mg in a 24 hours (h) urine sample (performed within 7 days prior to enrollment) * For patients \< 18 years of age: Serum creatinine =\< 1.5 x upper limit of normal (ULN), or measured or calculated creatinine clearance or radioisotope glomerular filtration rate (GFR) \>= 60 mL/min/1.73 m\^2 for patient with creatinine levels \> 1.5 x institutional ULN, or a serum creatinine based on age/gender as follows (performed within 7 days prior to enrollment): * 1 to \< 2 years - 0.6 mg/dL (male, female) * 2 to \< 6 years - 0.8 mg/dL (male, female) * 6 to \< 10 years - 1 mg/dL (male, female) * 10 to \< 13 years - 1.2 mg/dL (male, female) * 13 to \< 16 years - 1.5 mg/dL (male), 1.4 mg/dL (female) * \>= 16 years - 1.7 mg/dL (male), 1.4 mg/dL (female) * Creatinine clearance should be calculated per institutional standard * For patients \>= 18 years of age: Serum creatinine =\< 2 x ULN, or measured or calculated creatinine clearance or radioisotope GFR \>= 40 mL/min/1.73 m\^2 for patient with creatinine levels \> 2 x institutional ULN (performed within 7 days prior to enrollment) * Creatinine clearance should be calculated per institutional standard * Serum total bilirubin =\< 1.5 x ULN for age, or direct bilirubin =\< ULN for patients with total bilirubin levels \> 1.5 X ULN (performed within 7 days prior to enrollment) * Serum glutamic-oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) or serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) \< 3 x ULN for age (performed within 7 days prior to enrollment) * Albumin \> 2.5 mg/dL (performed within 7 days prior to enrollment) * Shortening fraction of \>= 27% by echocardiogram, or * Ejection fraction of \>= 50% by radionuclide angiogram * No history of myocardial infarction, severe or unstable angina, or peripheral vascular disease * Corrected QT (QTc) =\< 480 msec. Note: Patients with grade 1 prolonged QTc (450-480 msec) at the time of study enrollment should have correctable causes of prolonged QTc addressed if possible (i.e., electrolytes, medications) * International normalized ratio (INR) or prothrombin time (PT) =\< 1.5 X ULN. However, if patient is receiving anticoagulant therapy, PT or partial thromboplastin time (PTT) should be within therapeutic range of intended use of anticoagulants * Activated partial thromboplastin time (aPTT) =\< 1.5 X ULN unless patient is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants * A baseline blood pressure (BP) =\< the 95th percentile for age, height, and gender for patients \< 18 years old, or =\< 150 mmHg (systolic) and =\< 90 mmHg (diastolic) for patients \>= 18 years old * Note: 2 serial blood pressures should be taken at least 1 hour apart and averaged to determine baseline BP * Patients are eligible if on stable doses (\>= 7 days) of anti-hypertensive medications with a baseline BP meeting the criteria above Exclusion Criteria: * Patients unable to swallow whole tablets * Patients who in the opinion of the investigator are not able to comply with the study procedures are not eligible * Prior Therapy * Patients who have received prior therapy with axitinib, nivolumab, or other PD1/PD-L1 targeted therapies * Patients who have received prior therapy with more than one anti VEGF based agent (antibody or tyrosine kinase inhibitor) * Patients with hypersensitivity to axitinib, nivolumab, or any of its excipients * Patients who previously received an allogeneic stem cell transplant (SCT) or solid organ transplant are not eligible * Patients may not be receiving any other investigational agents (within 4 weeks prior to study enrollment) * Patients who have received prior anti-cancer monoclonal antibody (mAb) within 4 weeks prior to study enrollment or who have not recovered (i.e., =\< grade 1 or at baseline) from adverse events due to agents administered more than 4 weeks prior to enrollment * Surgery: Patients who have had or who are planning to have the following invasive procedures are not eligible: * Major surgical procedure, laparoscopic procedure, open biopsy, core biopsy, fine needle aspirate, or significant traumatic injury within 7 days prior to enrollment. NOTE: External central lines must be placed at least 3 days prior to planned treatment initiation and subcutaneous ports must be placed at least 7 days prior to planned treatment initiation * Patients who are planning cytoreductive surgery within the first 12 weeks following therapy initiation * Patients who have a serious or non-healing wound or ulcer at the time of study enrollment are not eligible * Patients who have a history of abdominal fistula, gastrointestinal perforation, or intra-abdominal abscess within 28 days of study enrollment are not eligible * Patients who have received prior targeted small molecule therapy within 2 weeks of enrollment or have not recovered (i.e., =\< grade 1 or at baseline) from adverse events due to agents administered more than 4 weeks prior to enrollment. NOTE: Subjects with =\< grade 2 neuropathy are an exception to this criterion and may qualify for the study * Pre-existing conditions, which may include: * Additional known malignancy that is progressing or requires active treatment. Exceptions include basal cell carcinoma of the skin, or squamous cell carcinoma of the skin that has undergone potentially curative therapy, or in situ cervical cancer * Patients with underlying immune deficiency, chronic infections including hepatitis, tuberculosis (TB), or autoimmune disease * Human immunodeficiency virus (HIV)-infected patients with the exception of patients on an effective anti-retroviral therapy with an undetectable viral load within 6 months prior to enrollment * Patients with underlying hematologic issues including congenital bleeding diathesis, known previous gastrointestinal (GI) bleeding requiring intervention within the past 6 months, history of hemoptysis within 42 days prior to study enrollment, active pulmonary emboli, or deep vein thromboses (DVT) that are not stable on anticoagulation regimen * Patients must not have had significant vascular disease (i.e. Moya-Moya, aortic aneurysm requiring surgical repair) * A known history of, or any evidence of active, non-infectious pneumonitis * Patients with known active central nervous system (CNS) metastases and/or carcinomatous meningitis or leptomeningeal disease. Patients with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging for at least 4 weeks prior to study enrollment and any neurologic symptoms have returned to baseline), have no evidence of new or enlarging brain metastases, and are not using steroids for at least 7 days prior to study enrollment. This exception does not include carcinomatous meningitis which is excluded regardless of clinical stability * Any uncontrolled, intercurrent illness including but not limited to ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia * Any serious medical or psychiatric illness/condition including substance use disorders likely in the judgment of the investigator(s) to interfere or limit compliance with study requirements/treatment * Patients with active autoimmune disease that has required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment * Treatments and/or medications the patient is receiving or has received that would make her/him ineligible, including: * Concomitant (or receipt of) treatment with medications that may affect the metabolism of nivolumab and/or axitinib within 7 days prior to planned first dose of protocol therapy * A live vaccine within 30 days of planned first dose of protocol therapy. NOTE: Inactivated flu vaccines are allowed; however intranasal influenza vaccines (e.g., Flu-Mist) are live attenuated vaccines, and are not allowed * Pregnancy and breast feeding * Due to risks of fetal and teratogenic adverse events as seen in animal studies, a negative pregnancy test must be obtained in females of childbearing potential, defined as females who are post-menarchal. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * Females of childbearing potential that are sexually active must agree to either practice 2 medically accepted highly-effective methods of contraception at the same time or abstain from heterosexual intercourse from the time of signing the informed consent through 5 months after the last dose of study drug * Lactating females are not eligible unless they have agreed not to breastfeed their infants starting with the first dose of study therapy through 5 months after the last dose of study therapy * Male patients of reproductive potential must agree to use an adequate method of contraception starting with the first dose of study therapy through 7 months after the last dose of study therapy. Prior history of vasectomy does not replace requirement for contraceptive use * Regulatory requirements * All patients and/or their parents or legal guardians must sign a written informed consent * All institutional, Food and Drug Administration (FDA), and National Cancer Institute (NCI) requirements for human studies must be met