So I have a Silver Husky on one character and a Black Bunny on the other and very often when changing maps I find that the Pet becomes unequipped. Also each time I log in I find that the Pet is unequipped also.
When changing maps, if you dont see your pet try hanging on a rope. Likely it got stuck in a corner and it's still 'out' but you can't see it. As for logging in, that's a known thing and should be changed in the new source.