Animals and caregivers?

Some caregivers won't mind at all, some may even be glad, some may not want the extra burden, it's all up to picking the right caregiver .

DH aunt had aides took care of her dog along with her. Someone who wasn’t willing to accept the dog as a positive part of her life wasn’t a good fit for her.
She had one person who applied who passed on the opportunity due to her dog. That was fine. Not everyone is the “right” one.

What I always did with clients who had dogs was insist that the dogs be confined to another room or (weather permitting) kept outside during my shift at the client's home.

I never objected to meeting a pup and letting them get used to me if someone other than the client was in the house and they had them on a leash and control of it. If there wasn't someone there then the dogs had to stay confined or outside.
Many dogs over the years got familar with me and they weren't a problem. So long as they didn't interfere with care.

The individual aide usually decides his/her comfort level with pets.