Dunno about anyone else but 'just look in the archive' is getting to be rather difficult.
We get an average of 60 new pets each month. The archive now has ten years to flip through. That's not a small chunk of time anymore.
Are we really expecting people nowadays to 'just look through the archives"? The thousands and thousands of pets there are on this game? Not sure about you but this was fine in 2010. Now it's just unmanageable and unrealistic to expect players to just know the ballpark of every pet in the archive.
I have a pet that's mixed in with my monthly adopts. I go to trade someone and trade some butterfly wolves away. Whoops, those were this months store pets that I bought at the beginning of the month but haven't looked at them in three weeks. True story.
The fact that pets don't get their rarities for 3 months+ nowadays compounds this issue.
Look! Here's a dog.

Let's pretend it belongs to CS and I'm a new player. I got this dog from someone's give away. We have NO idea where it belongs in the archives but it has an adoption date of Dec 14th 2016. So is it a regular adoptable? *scrolls through pets*
Nope.
Was it a Christmas pet? *scrolls through 4 pages*
Nope.
Well, if I'm lucky and the rarity does show. (Today it would not! Dec '16 store pets don't have rarities!), It would most likely say VR.
And of course we all know that a vr totally equals a vr so this vr (a store pet) equals a vr event pet from two years ago.
I could ask someone but my knowledgeable friend is not online. I'm sure it's fine this once because that user wants a response right now or they might cancel the trade with one of my dreamies in it! It looks fine to me. I'd better hit accept before they change their mind. How valuable can this thing be if I got it in a giveaway?
I think at the very least, they need a marker that ids them as a store pet, something obtained not by regular adoption. We have that marker on BBS pets, why not one for store pets?