No, SARS-CoV-2 acutely depresses total lymphocyte count in a subset of patients. This includes both CD4 and CD8 T cells. The mechanism of this decrease is unclear, but transient lymphopenia is not uncommon in severe infection of just about any cause, including bacteria and other viruses.
HIV infects macrophages and CD4+ lymphocytes directly, and gradually kills them off over a period of years. In contrast, CD8+ cells increase in early infection, but tank in later disease. Concomitantly, B cells are activated and there is a paradoxical hypergammaglobulinemia, but humoral immune response is decreased. "Cytokine storm" is not a feature of HIV disease.
AFIK, the only commonality is decease in lymphocyte count, but the mechanism, duration and extent of immune involvement of the two viruses are otherwise quite different. And to be clear, a lot of infections transiently decrease lymphocyte counts.
What other common characteristics does SARS-CoV-2 share with HIV? I know the the drug you're promoting targets CCR5, which is a co-receptor for HIV. There were early reports that SARS-CoV-2 also involves that receptor, but I thought they were debunked.
Though I've not seen this comparison made, I was thinking SARS-CoV-2 immunology was potentially more like dengue. Dengue causes decreases in cell counts, and has a hyperimmune phenomenon like the "cytokine storm" described in COVID-19. This results from re-infection with a second dengue strain, due to cross reacting antibodies - is it possible people who experience the "cytokine storm" have an overzealous immune response from prior, non-COVID coronavirus infection?