As far as I understand it, AIDS is a syndrome rather than a disease because while the cause of the immune system deficiency has a single cause, many of the symptoms are caused by secondary problems that arise because of the immune system deficiency, and so the symptoms do not have a single direct cause, even though they're all indirectly the result of HIV, and the immune deficiency itself is a direct result of HIV
I think it would be more accurate to say that AIDS is a syndrome because it's defined as such. You can also speak of HIV disease[1]. HIV disease would be the disease associated with the human immunodeficiency virus. AIDS refers to a particular syndrome. They mostly overlap, but they describe slightly different concepts. My speculation is that the reason the distinction persists is mainly historical: AIDS was described and became well known before the HIV virus was discovered and was demonstrated as the causative agent.
> My speculation is that the reason the distinction persists is mainly historical ....
I imagine another reason for the distinction is that AIDS is typically a late stage of HIV infection. That is, people who develop AIDS have usually been infected with HIV for some time.