About autopotting and different pets

My guess would be that it has something to do with classes and skills and not the actual pet itself.

I think so too.

Unless Nexon is even stupider than I give them credit for, the code for auto-pot is linked to the equip (Auto-HP Pouch), not the pet. And even if it is per pet, it would be copy-pasted.

However, auto-pot is known to fail when you are potting manually, buffing, or looting. It might be that Mechanics' skills/summons/mount/etc interfere with auto-pot relatively frequently.