Atezolizumab in Combination With a Multi-Kinase Inhibitor for the Treatment of Unresectable, Locally Advanced, or Metastatic Liver Cancer

Inclusion Criteria: * Provide written informed consent =\< 28 days prior to randomization * Willing to return to enrolling institution for follow-up (during the Active Monitoring Phase of the study) * NOTE: During the Active Monitoring Phase of a study (i.e., active treatment and clinical follow-up), participants must be willing to return to the consenting institution for follow-up * Age \>= 18 years * Hepatocellular carcinoma (HCC) confirmed by histological/cytological diagnosis or clinically per the American Association for the Study of Liver Diseases (AASLD) or WASL 2018 criteria * Locally advanced, metastatic and/or unresectable disease that is not amendable to curative treatment * Previously progressed on atezolizumab in combination with bevacizumab as first line systemic therapy for advanced disease * NOTE: 2nd line patients only * Eastern Cooperative Oncology Group (ECOG) Performance Status 0 or 1 * Child Pugh class A * Documented virology status of hepatitis, as confirmed by screening hepatitis B virus (HBV) and hepatitis C virus (HCV) serology tests. * For subjects with active HBV, HBV deoxyribonucleic acid (DNA) \< 500 IU/mL obtained ≤ =\< 28 days prior to randomization, and anti-HBV treatment (per local standard of care; e.g., entecavir) for a minimum of 14 days prior to randomization and willingness to continue treatment for the length of the study * At least one measurable untreated malignant lesion per RECIST v1.1. Subjects who previously received local therapy (e.g., ablation, percutaneous ethanol injection, trans-arterial embolization/chemo-embolization) are eligible provided the target lesion(s) have not been previously treated with local therapy or the target lesion(s) within the field of local therapy have subsequently progressed in accordance with RECIST v1.1 * Consent to using archival tumor tissues, if available * NOTE: Non-availability of tumor tissue does not exclude the subject. * Willingness to provide mandatory blood specimens for correlative research * Willingness to provide mandatory tissue specimens for correlative research for the first 10 patients per arm (Mayo Clinic Rochester and Mayo Clinic Arizona ONLY) * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9/L (1500/uL) without granulocyte colony-stimulating factor support (obtained =\< 28 days prior to randomization) * Lymphocyte count \>= 0.5 x 10\^9/L (500/uL) (obtained =\< 28 days prior to randomization) * Platelet count \>= 75 x 10\^9/L (75,000/uL) (obtained =\< 28 days prior to randomization) * Hemoglobin \>= 90 g/L (9 g/dL) (obtained =\< 28 days prior to randomization) * Subjects may be transfused to meet this criterion * Aspartate aminotransferase (AST), alanine aminotransferase (ALT), and alkaline phosphatase (ALP) =\< 5 x upper limit of normal (ULN) (obtained =\< 28 days prior to randomization) * Total bilirubin =\< 3 x ULN (obtained =\< 28 days prior to randomization) * Serum albumin \>= 30 g/L (3.0 g/dL) (obtained =\< 28 days prior to randomization) * For subjects not receiving therapeutic anticoagulation: international normalized ratio (INR) or partial thromboplastin time (aPTT) =\< 1.5 × ULN (obtained =\< 28 days prior to randomization) * Serum creatinine =\< 2 x ULN or creatinine clearance \>= 30 mL/min (calculated using the Cockcroft-Gault formula) (obtained =\< 28 days prior to randomization) * Negative pregnancy test done =\< 14 days prior to randomization, for women of childbearing potential only * NOTE: If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * Resolution of any acute, clinically significant treatment-related toxicity from prior therapy to grade =\< 1 prior to randomization, with the exception of alopecia and peripheral sensory neuropathy. * Subjects of childbearing potential agree to use two forms of medically approved contraception while taking the study drug and for at least 5 months after the last dose of atezolizumab or multi-kinase inhibitor. Subjects with partners of childbearing potential agree to use condoms, even after vasectomy, to avoid potential drug exposure to partner during study drug and for 5 months following the last dose of study drug * Ability to take oral medications Exclusion Criteria: * Known diagnosis of fibrolamellar carcinoma, sarcomatoid carcinoma or mixed hepatocellular cholangiocarcinoma * Prior multi-kinase inhibitor treatment for advanced disease (e.g., cabozantinib, lenvatinib, sorafenib, regorafenib) * NOTE: Use of multi-kinase inhibitor(s) for adjuvant or as part of loco-regional therapies is allowed as long as the therapy was completed \>= 6 months prior to randomization * Any of the following prior therapies: * Major surgery =\< 4 weeks prior to randomization; Minor surgery =\< 7 days prior to randomization (e.g., simple excision, tooth extraction, insertion of central lines/Mediport). Subjects with clinically relevant complications from prior surgery are not eligible * Any anti-cancer agent =\< 2 weeks prior to randomization * Radiation therapy =\< 4 weeks (1 week for palliative radiation for bone metastases and/or for pain control) or radionuclide treatment (e.g., I-131 or Y-90) =\< 6 weeks prior to randomization * Treatment with investigational therapy =\< 28 days prior to randomization * Known brain or leptomeningeal metastasis * Known co-infection of HBV and HCV. Subjects with a history of HCV infection but who are negative for HCV ribonucleic acid (RNA) by polymerase chain reaction (PCR) will be considered non-infected with HCV * Active or history of autoimmune disease or immune deficiency, including, but not limited to, myasthenia gravis, myositis, autoimmune hepatitis, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, antiphospholipid antibody syndrome, Wegener granulomatosis, Sjogren syndrome, Guillain-Barre syndrome, or multiple sclerosis with the following exceptions: * Subjects with a history of autoimmune-relate