If someone has a dog or a bird, I'm in from the start. Shows they aren't bending over backwards to give themselves something wacky or unique, this is likely a laid-back player who will be a reliable team member. Maybe not, but it's not a red flag.
If someone says they have a wolverine (which functions identically to a dog but it's a quirkier animal) or a cat (but it has batwings and flaps around like a mini Toothless from How To Train Your Dragon and they describe it comedically reacting to everything that happens in great detail by halting the game to play HtTYD memes clips and going "that's him right now") then I need to be won over and convinced this isn't going to be annoying. I've had unorthodox animal companions turn out fine or even great, but it's that initial mindset of "how can I make my character more aesthetically special than a common person" that is often a red flag.
There's a character in my current campaign who has a little skeleton cat that runs around. But it acts just like a normal cat, doesn't have a stupid name, and his character's culture revolves around honoring the dead and performing necromancy as funerary rights. So it doesn't really stand out in an obnoxious way, he didn't give himself this cat so he'd have a shiny toy to add to his HeroForge mini, it blends in with the setting and adds more depth to his character. Bonus, it's insanely useful in gameplay, as he can have it deliver healing items and scout ahead for better visibility.
But if the cat was named "Spoopy McSkeletonpants", had no connection to his character's backstory or culture, and was just an excuse to make skeleton puns, it would clash really badly with the more serious elements of the story and serve no purpose beyond being annoying. A lot of unorthodox animal companions have that kind of theater kid "look at me" energy.