It's already marked as shop=pet_grooming. Also the documentation for https://wiki.openstreetmap.org/wiki/Key:animal (animal=wellness) says, "Animal wellness centre (e.g. dog groomer).
Primarily dogs are getting coiffures, but there might be also services like clipping claws/cutting nails or other beautification services, so we chose the term 'wellness'."

The documentation that you listed doesn't mention that  animal=wellness is deprecated or discouraged. It just discourages automated edits, and that it's a possible synonym.

I see no mention of which tag is preferable.

So according to the current documentation, this is properly tagged. It has both tags.

If you feel otherwise, then the documentation should be updated to clarify which tags are deprecated.