Do I really need 10 AWG wire for a AC condenser that says 15 AMP max?
It says specifically that you must use a 15A breaker max.  Max means not 20.  

It also says you must use a HACR breaker.   All new breakers are HACR.  

You are required to comply with a component's labeling and instructions. NEC 110.3.   You cannot go "la la la, I think I'll do something else because reasons".  Unless you can convince your AHJ of the reasons.  (AHJ is your local government which issues permits and does inspections.)

Why is the wire 10 AWG? Being Future-proof.  The cost of wire ($5) is next to nothing compared to the cost of tearing out 12 AWG wire to replace it with 10 AWG (hundred$). The last guy installed the heavier wire so you'd be "good to go".   You do still need to change the breaker to correct size, but that's cheap.