I'll tell you why it is not only appropriate but necessary.

Because by calling our rescue pets "rescue pets" we promote the idea of adopting pets from rescue organizations.

That is good PR for animal rescue organizations because it tells people what great pets rescue animals make.

That encourages people to donate money to rescue organizations and to adopt rescue pets of their own.

You can't get better advertising/PR than in-person recommendations.

People who meet your rescue animal and admire it will at least start thinking about adopting their own rescue pet some time in the future. If not immediately!

In fact, I'm confident that one or more of your adopters has probably encouraged another person to get their own rescue animal from you.

We actually HELP rescue organizations by talking about our pets as rescues.

Me, I've actually promoted getting older rescue animals. They may be hard to find homes for, but they make the very best pets. I've adopted two rescue kitties (in succession) who were each 12 years old when I got them. My first rescue kitty (who passed at 18) was the very best, most loving kitty I've ever had in my 73 years of having an animal in the house. I miss her dearly.

The second one I got two years ago is differently wonderful, and is actually quite kittenish still at 14 years old. A kitty properly cared for can live up to 25 years.