Patients receiving immunosuppressive drugs do fall into the same category as patients with HIV-related acquired immunodeficiency - that is, they are all immunocompromised and face similar risks of infection, cancer, etc. The difference emerges over time: when the immunosuppression is controlled and monitored, a patient can retain some immune function, and a physician can help a patient adjust and rapidly respond to opportunistic infections. Drugs can be given in calculated doses; viral infections which disable the body's ability to stop viral infections typically only get worse. To elaborate:
HIV causes immunodeficiency by killing T cells. Firstly, it preferentially infects CD4+ T cells. This process can trigger an inflammatory response and (1) kill the infected T cell, as well as (2) bystander cells. 225
Once the virus has successfully infected a CD4+ T cell, it can happily produce more viruses without killing the cell, which is how the infection can spread - or it can (3) rupture the cell membrane and kill the T cell. Finally, CD8+ T cells can notice that CD4+ T cells are getting infected by a virus and start (4) killing CD4+ T cells.
Immunosuppression after organ transplant can be more specific. It is typically achieved via a combination of drugs which can (1) reduce immune cell activation, (2) slow down immune cell proliferation, or (3) block the activating receptors on T or B cells. The amount and type of suppressants depend on the phase of treatment: induction of immunosuppression, maintenance, or treatment of transplant rejection.1 These drugs are also often administered alongside preventative antimicrobial drugs.
The immunosuppressants which are administered during the maintenance phase - i.e. for the rest of the patient's life - are often the drugs which dampen immune activation instead of eradicating all T cells, and are given at much lower doses. For example, calcineurin inhibitors block the [calcineurin-NFAT pathway];2 when this pathway is blocked, production of immune activators like IL-2 is reduced, and reduces T cell function.
TL;DR: "induced AIDS" is milder and more controlled than actual AIDS, and doctors work very hard to keep patients within a safe range of immune function/suppression.
PS: Good question! Made me think :) I've written some papers on organ transplantation before so am happy to do some digging to add sources. 225