With us it depends on the name. For example, our first Singapura, who came from a breeder, was named “Tenshi” when we got him. We liked the name and it seemed to fit him, so we kept it. Our second Singapura was named “Max” (his siblings were “Minnie” and “Eminem” or “M&M.”) We didn’t like that one, so we renamed him “Ozzy.”
Our two Russian Blues came with the names “Aria Bella” and “Sonata” (the whole litter was named after musical terms). We liked the names but decided to rename Aria Bella to “Grace” after a character in a book we liked, but we kept Sonata (who we now call “Fred”). Grace, who we call “Boo” most of the time, often gets referred to as “Graciela Aria Bella Boo,” so we kinda-sorta kept her name.
Our three rescue tabbies didn’t come with names, so we just named them ourselves (Clarice, Nabby, and Tessa (aka “Bug”)).
Our pet did not respond to the name the shelter gave her. We tried it, but it didn’t really suit her. Eventually (after a coupe of weeks) she started responding to a couple of nicknames we were using, so the three of us sort of agreed on a new name for her.
Maybe, maybe not. Shamus the Maine coon is now six, and was 18 months old when we got him. His old name was Otis. He started responding to his new name within days, but still will look up if you say “Otis,” so I think he at least is capable of knowing that names refer to him (and “Oogie Boogie” too, which only I call him).
My current dog is a rescue dog. He had been abused and starved. Then he was fostered for a while. He came to us with the name “Rollo.” i didn’t know if that was his foster name or his abuse name. Either way, I decided he was geting a new start with us, so he could have a new name, “Dusty” as well. He adapted to his new name with no problem. It’s taken about four years, but he is really starting to bloom.
All of my cats have been rescues. One came to me with the tentative name of “Stinkerbelle” which I determined had to go. I took a good look at him and renamed him Frank. Sometimes he did indeed stink but he always appreciated his bath when he developed an odor (a rare thing for a cat).
Two of my rescues retained the names they came with because they recognized them. The others were young and either didn’t have names or didn’t recognize their initial names.
My cat was named by her previous owner, and it fit her really well (Olive, for her eyes) so I kept the name. Although I’m not sure if she knows what her name is - once I idly called her “Orange” and she still came over looking for a Greenie.
My most recent cat came from the shelter, and I had already thought of a name before I went to pick out a new cat. When I got her, and was filling out the paperwork, I saw that the temporary name they had given her was so good, I kept that, instead of naming her what I had previously come up with, so the answer to the OP is: It depends on the name the cat already has.
I probably wouldn’t rename a dog, as they actually respond to their name. But we renamed our cat, from Sandy to the more appropriate and dignified Wowbagger. The transition went off without a hitch.
My dog was called Chica at the shelter, and we changed in immediately to Dixie. I figure that the name Chica, if she ever recognized it at all, probably reminds her of a dark time in her life.
All or most of mine have been strays that have either accosted me in the street, or have been given to me. A couple of them have been renamed.
‘Hans,’ a German Shepard we got long ago through a local vet’s office, started out as ‘Rommel.’ Mother didn’t like the name and promptly re-christened him ‘Hans.’
Some of you will remember here on the Dope when my beloved Murphy passed away several years ago and I learned of a tiny kitten down in Georgia named ‘Speck,’ who needed a home.
She came to live with me shortly afterwards (Hi, CM, wherever you are - and many, many thanks. She’s still as awesome as the day I got her!!).
On the ride home to Indiana from Tennessee (my girl posse and I met up with CM and her hubby in Chattanooga to get Speck), Speck was renamed Magnolia, later shortened to Maggie.
Miss Widget, currently gorked out in my lap, was barely four weeks old when I got her from my niece. She just looked like a Widget.
