Is a homeless person's pet also homeless?

I just read in the paper that some vets did some volunteer work to help people who brought their pets to Dolores park. The headline read "Vets who fret for homeless pets". To me, it sounds like the pets are homeless. I would say they are not. My (limited) experience is that dogs and cats think of "home" as wherever they get their food. If a homeless person is giving a dog food, then that pet is not homeless, its home is with the homeless person.

While I'm on the homeless subject, I've always thought that there is no scenario under which giving money to someone on the street will help them get off the street. This does not apply to street performers (Bush Man makes $60K/year), just people that beg. Does anyone know about this subject? Is it possible for them to get off the street from direct giving?