Reasons to collect more than one of any given pet:
- Some pets have abilities that benefit noticeably from multiple breeds (i.e. I run a P/P Flayer Youngling, some strategies call for a S/S).
- Some pets are pretty OP (and have unique movesets) that you may want multiples of it (i.e. Ikky, I know a few people that run 3 Ikky teams for PVE battles, as easymode).
- Some pets have multiple skins/sizes, so for someone that values that, you would need to have up to 3 of one pet (i.e. Parrot has 3 unique colors).
Reasons to not collect multiples of any given pet:
- As you noted the journal space is limited and it's impossible to have 3 of every pet, so as new pets come out you'll have to keep releasing extra pets to make room.
- If you're OCD it'll drive you crazy to see all those pets! And then there's the whole issue of stoning and leveling them!

And, honestly, are you really going to use them all?
I think most players do keep extras of pets that they find particularly useful. I collect for aesthetics so the duplicate pets I have are there to cover unique looks.