Do the pets become strays that someone else can tame or do they just get deleted (unless the person who released them pays the fee to get them back)?

Edit: I've also seen it's frowned upon to release pets so when is it socially okay in this game to release a pet?

Yeah like you said, they stay released until reclaimed by the person who released them! Not exactly deleted, just in a state of stasis, nobody else can claim them.

It's not really frowned upon, some people simply don't like it when their pets get purchased just to be released. In fact, everyone that does breeding projects will likely resort to releasing, as it's simply not sustainable to keep every pet if you breed hundreds of them. That's why the game even encourages releasing (during spring fest and patriotic fest especially)!

Oh okay that makes sense, thanks for the info