Ginger is “Ginchy” or “Buddy.”
Rascal is occasionally addressed as “old man”, since he will be turning 16 next April which makes him older than I am in cat years. I will also refer to him as the bane of my existence when he wakes me up at 5am.
Toby, T bone bubba, bubba guts, big T, tobymory, killah, my boo, butter lover. And many more.
Not really a cat person. But I love all animals.
One of our dogs that we adopted was named ‘Buddy’. We changed his name to Jackson or Jax. I do not call him ‘Buddy’ (though he really is my buddy, he loves me). I don’t want him to re-live what every happened to him as a shelter/foster dog by calling him ‘Buddy’.
Instead of ‘Buddy’, I simply say his new name, Jax, or ‘My Friend’. Or ‘Big guy’.
Not sure if this counts for the purposes of this thread, since she was only ever called by her full name, but I used to sing a variation of “My Sharona” to one of my cats:
Ooh, my little pretty one,
My kitty one,
Whatcha gonna do with your time … Fiona?
I could substitute almost anything for “time,” depending on what she was doing: “food,” “toy,” and so on. She knew her name, and would usually respond to the song by looking at me and saying, “Meow?”
Or sometimes cuddlepuss, when he curls up on my lap. Merry is also called cuddlepuss, when he begs to be lifted up. I cradle him in my arm, like a baby, and tickle his tummy and call him cuddlepuss.
They are cuddly in such different ways, yet somehow, they both get the same name.
Kitty, catface, cathead, catbutt, Cyggy, meow-meow, smooth kitty, black kitty, mean kitty, sleepy kitty, hungry kitty, mega kitty, flat kitty, snugglebottom, blorpo, bleep-blop, jerkbutt, hypercat, bitey, scratchy, Cygnus.
My Cat’s real name; “Garbage” but his REAL name; “Garby”. If you’ve watched ‘The Office’ Dwight calls a cat Garbage because he found him near the dumpster. Guess where I found Garby?
Garby, Sadly passed.
This. Usually “kitty”, but sometimes “stupid cat”.
(Full disclosure, I am not a cat owner, but we’ve often had cats in the house when kids/grandkids wind up living with us temporarily. I can’t be bothered to learn their names, and they don’t listen to me anyway. 
)
Well, if we’re moving beyond cats for a moment, my Quarter Horse was registered as Skip in Place, but his barn name was Nick or Nicky, sometimes Pickle, and for full dramatic effect The Magnificent Pickle.
My Bitch-er-Ella is called Ella. She’s a lousy pet (she has PTSD and is probably bi-polar - can’t be groomed or picked up), but very beautiful. I’m stuck with her, but she has moments when she is delighted to see me and then I call her ‘Purr Cat’. When she is vocal, I call her ‘Meow Cat’. We watch birds at the feeder where she chatters - when I spot a fat juicy bird hopping around outside, I only have to say ‘Bird?’ and she chitters back at me. Maybe I should call her Birdie.
I forgot that I tend to call all black cats Boo-berry. My cats or yours.
I tend to call all dogs “pup” or “puppydog”. Even if they’re full size aged Great Danes.
Confuses some humans. Doesn’t seem to bother the dogs any.
It always amuses me that prize-winning dogs (and cats?) have a fancy name for show use, but usually a mundane name at home. One I found was Whilby’s Ungulate Pinbill (answers to “JimJim”)
Sometimes I’ll bust a rhyme rapping to my cat, freestyle for the funky feline he’s all that my cat, the killah, the butter lover he’s big Tbone with the big guts just a bubba dog in cat suit meow.
I’ve embarrassed him endlessly. Me? Idgaf what that scary neighbor with rottie thinks his dog loves my cats Scary not so much.
Rocket J. Squirrel: “sweet boy” “perfect angel”, “Rocky” and other variations.
Juno: “dammit kitten” (she’s nearly 4 and still gets called kitten). “sweet baby” “good girl” “Holy f**kin crap where did you come from” and “could I have a little of my bed please, demon?”
Also I think often true of thoroughbred race horses, who have to be formally named according to a strict set of rules.
If you register your Thoroughbred with the (US) Jockey Club, yes, there are rules. Or you can register a foal without an assigned name. My own late TB (never raced, bred for riding) had no official Jockey Club registered name, just a barn name of Ben, Never Ben Better for showing.
it’s in fact very common for horses with competition careers to have a fancy official name and an informal barn name.
Ella, Smella, Cinderella. Ella, short for Bitcherella, she is a nasty piece of work I was gifted and stuck with. Has PTSD and is bipolar. Extremely beautiful Maine Coon. The only word she understands is ‘bird’ - when I spot birds at the feeder, I say, ‘Bird! Come see the Bird!’ and she comes running, chittering and whinging.
We have Rudy and Matilda. Rudy used to be called “Fat Boy” until recently, and usually “the pain” and “dummy.” Tilly is “pretty girl” and “too damned smart.” Though I like to point at them and say “You are a cat!” once in a while.