From an out of universe point-of-view, it makes complete sense. Pets are not really essential to the story. In-universe, there are clearly enough to keep a reasonably-sized vet open, so we probably just haven't seen them yet.

Somebody's got a pet frogmoth.

I think you're right that it's a budgetary issue.  With this giant cast, something had to give, and not involving a lot of animals and their trainers makes room for more humans to get paid and keeps the production schedule moving along.  I'm reminded of the difficult kitten scene in Truffaut's Day for Night. 

I think Carl's got a pet Pine Weasel, which answers to his whistle & doesn't like to be disturbed in the mornings...

After 9:30am. NEVER BEFORE! 🙂