So, max is twenty pets, when used, does it add more than twenty?

Yes. Those boxes are the only way (at least so far) to get more than 20.

you could also get more than 20 if you had 20 when the plushies were changed before, but yeah it's the only way to get over 20 currently.

^^ Thank you tons! I will save them then.

make sure to lock them in your inventory, so they don't get eaten or zapper or something!

Oh, good point. Thanks for the tip!