I adopted a pet from Adopt R Us (which, btw., has been extremely lowly populated to empty for a long time). I checked to make sure it was reasonably old. When it arrived, I had to find out, the profile lacked "Legacy Name". Sad, but no big deal, I'll just put it back... BUT "It costs 10,000 sP to abandon a pet." Not necessarily a fortune, if you check 1 pet, but if that goes into masses...
There should be some statement about legacy or not on the overall pet info in the pound.
you can always check their profile before adopting them by copy-pasting the pet ID into the end of this URL: https://subeta.net/petinfo.php?petid=