To clarify: you will not use a 1 3/8 course thread when you have a 1 1/2 fine thread drain. Use what matches the drain you pulled out. It's likely your hardware store will sell a drain that fits both 1 3/8 and 1 1/2. This means it has an adapter that will let you use it on either size drain.
Yes, you want to use the same size/thread drain that came out. If you aren't sure what that is, take the old one into a hardware store and match it up there.
I was wondering the same thing. I thought there was gonna be a "Gotta find a place that has that size.." or something....Don't understand how a larger one can fit...