Do tub drains of the same diameter ever have different thread-per-inch counts?
I have never come across flanges that weren't compatible, and haven't seen manufacturers specify TPI on their kits or replacement parts.  There are plenty of oddball pieces here and there but upon coming across those (on rare occasions), I'd be replacing the entire assembly.

There is really no way to know the TPI without removing the flange, but chances are you will be dealing with coarse threads.