Affixes? Suffixes?
This is a thing on other sites i have been on such as Howrse, where you can create breeding groups for your pets and add little things like 'of the Magical Woods' to the ends of their names. You could restrict these groups to a certain themed or keep it open to all of your pets from one species but the idea is that you can then create little groups within your pets.
This would majorly help with breeding as you could then filter through only pets in a certain breeding group to find the one you are looking for!

(On other sites there was also an option to create shared breeding groups with multiple people so that there could be free breeding between peoples pets if you are doing a project together without having to manually set up each individual pet for a certain person)

One thing worthy of note, on Howrse (at least back when I had an account), affixes could only be changed by the horse's original owner. So if you bought a horse with an affix, it would still be a part of that breeding group and the original owner could easily be found. Of course, on Sylestia, the original owner is just shown directly on the pet's page. Affixes would be cool for the reasons mentioned above, though.

True! In this case then it would mainly be for breeding purposes or collections i guess (i.e. if you click on the affix then you can see all the pets with that affix)