I suggest that the name disappear, just like with foresting or shrining!
I would assume all the identifying info would disappear as well, like id, intuition stats, effects, etc.
I think it's a fair tradeoff, honestly, if you want to turn a creatu into a little lunchbox version of themselves. It's easy to slap a plush on a creatu that already has all the stuff you want, I think! (at least for now?)
Though, of course, this is why a feature like this should be behind a short quest or something that takes a bit of effort so that no accidents happen.
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Hehe, the vibes were in the air
But yes, I would go crazy for just having a bunch of easy pets to sell or just have in my gallery~