What does it mean when a pets name is in purple?

Example:
https://www.sylestia.com/view/pets/?petid=7255806

it just means its a specially obtained pet that was given that name when it was created.
the one you sent specifically was a shell pet obtained from combining sea shells during the summer festival; people usually like them because sea shell pets specifically can have special traits on them (on the ryori its the Arcanist Wings, its a lost grove trait). :)