You can “adopt” a new name, you can “adopt” a sarcastic tone, you can “adopt” a highway.
Words have meanings beyond whatever it is you happen to use them for, and it’s not “creepy” for people to use those meanings. Adopting an animal makes perfect sense and isn’t creepy in the slightest unless you’re the sort of person who can’t separate a word from the context where they first heard it, or can’t understand that context matters.
Another less anthropomorphic way of looking at this - plate repositioning dog doesn’t want it’s back to his fellow diner; takes up a more strategic position - slightly behind the other (in sight-line).
Alpha dogs will eat all it wants; the subordinate dog will make way for the alpha until they’re done.
Hard to say which one of these two is the alpha.
Separated eating locations seems to indicate the server is addressing a previous problem.
Some dogs truly are rescued. I had one that was saved from a rural puppy mill that was raided by the local authorities. She spent six years living outdoors in a wire cage having one litter after another. She was a broken spirit but she was the sweetest girl you could imagine. The remaining years of her life were surrounded by unconditional love and warmth. I still miss her like crazy.