no, AIDS is one of multiple opportunistic viruses that a non-HIV infected person would generally not be susceptible to. But yes, you could get the HIV virus from a cat, if the cat had the HIV virus and you drank its blood, or shared needles with it or something along that lines.
It is theorized that the HIV virus was spread to humans by monkeys, more specifically they were being slaughtered, had some form of HIV and the humans injested the blood somehow and got the virus. (Run on sentence)