Charging a AA recharable battery
I know that there are modules like the TP4056 for Lipo batteries and 18650 cells, but I don't know that there is something similar for an AA battery.

it's not about the form factor (AA), but about the chemistry (NiMH/NiCd?).
Yes, charge ICs for such batteries are available. You'd want to check the offerings of the major power IC manufacturers (Maxim, Texas Instruments, On Semi, NXP,…).

Can I just connect the battery to 1.2V and limit the current to 150ma?

That voltage will be too low to fully charge, so the answer to that question is no.
Usually, when building a &quot;dumb&quot; charger (not a fast charger), you'd enforce a constant current (typically: (battery capacity)/(10 hours)) and just charge for the necessary time.

At what voltage is the battery fully charged?

Doesn't really work like that for NiMH batteries; the terminal voltage depends on the aging state of the battery, kind of. It'll be at least something like 1.41 V, but it really depends.
All in all, NiMH and LiPo and Lead acid batteries need different charging approaches. There's really a lot of (older, because nearly obsolete) application  notes by aforementioned IC manufacturers that you can read, which would be better than anyone could write in a short answer. Read them.