So I was thinking maybe instead of releasing pets, you might be able to "recycle" them by getting coins in trade for your unwanted pets.

That's what selling pets is. Going for a recycle option would lead to multiple things, namely:
- Less interactions between players, as pets are more easily discarded then being sold to other players.
- Overpriced pets. The only pets that will be sold will be worth much more than the recycle price gives you, making buying from other player's less favourable.
- They will have a less 'pet' feel to it and more like an 'item' feel, just like you can sell normal gear and items.

Actually, I believe what she means is that the player is given a small amount of gold, (less than 20g, probably), when they release a pet. This gives releasing more of a positive reinforcement and would help many players who loathe to release their pets (keeping their stables open). And pets over that amount would not be overpriced- my opinion only, obviously. Releasing pets is better than sending them to an owner who may not feed them.

And there is another thread addressing this subject: https://www.sylestia.com/forums/?thread=50154