If my apartment building just got sold and the new owner wants to get rid of my dogs, can they legally do that?

It depends on the terms of your lease if it is for a fixed term. If it is month to month, then the terms can change each month. "Assisted Living" is not a protected class under the ADA. Under California law an Emotional Support animal may confer some rights.

The new buyer steps into the shoes of the old owner. Until there is a new lease [in a month to month tenancy it automatically renews, but the owner can offer new terms if gives 30 day notice [60 if lived there at least one year], the exact terms of the old one continue to apply, and whatever knowledge of the prior owner and their agents [building manager] had is assumed to be part of what the new owner knows. I do not know anything about assisted living animals, but i assume in California they probably can not be part of any exclusion of pets clause in a lease, so you should be able to keep the one already certified [the company that does the certification will know what your local laws are, as individual cities, especially if they have rent control, may have additional tenant friendly rules]. Ask the manager why the new owner wants the dogs out, considering all the circumstances that exist. Does he just want to add a pet charge, worry about apartment damage, clean up outside your actual unit but on the common grounds, etc. Most insurance companies as to the building owner allow tenant pets except for a few large, very aggressive types.