While not exactly recession-proof, I would say it could be fairly resilient in the more advanced economies compared to say, some other minor luxuries. As already stated, people there tend to budget with pets in mind.
In more developing economies, probably not as much resilient. It would require that it at least have a large middle class that can afford the services/products. For example my cat had some problems giving birth (first time I actually had a cat had problems like this) and we asked around and apparently these are expensive - just as expensive as human procedures sometimes.
So yes, we can afford the food and litter so those manufacturers get a bit of our lower income money but not the vets, who would be relying on the middle class for their income.
I don’t agree that the pet industry is recession proof. Shelters around me are already asking people to adopt due to animals being dropped off. People are leaving their pets as costs rise and they’re forced to deal with difficult financial choices. While pets have become part of the family, it would seem they are expendable in tough times.
Sadly that's the case. Most people would rather give up their pets than let their children starve.