The pet store and its related items are indeed only supposed to be granted to players who had purchased the limited edition. They are not meant to be registration gifts for the regular EP. The registration gift for the regular Pets EP is a set of two items that includes a futuristic pet bed and food bowl, not the pet store. I have a related question that I will ask by PM.
The Neighborhood Pet Adoption Center scenario is very prone to breaking the game. That is why NRaas Overwatch offers an options to always shut it down once it is noticed and NRaas StoryProgression does not allow the scenario to ever begin with no options being offered. On Overwatch, the setting to use is NRaas > Overwatch > Settings > Stop Pet Adoption > True (this means the Adoption Center scenario, not pet adoptions in general from other sources as those are fine).
Edit: Okay, the pet store part is all sorted out. The player purchased Pets recently by way of Origin and the pet store landed in their purchase history. So by the official rules they were not really entitled to it since the store was content that only came with the Limited Edition release of Pets, but it was granted to them anyway. When things like this happen, the players are indeed entitled to the content, if they can get it working and keep it so. Like many other players from a glitched correction EA made a very long time ago, I have a free copy of Barnacle Bay that I never asked for and was never officially supposed to be entitled to. :)