No. She’s a rescue cat, and was previously semi-feral, and is unusually small and has a couple of kittenish features even five years later, so they might have underestimated her age. Still, as long as it’s within a year it’s good enough for veterinary purposes.
If and when I get a dog I might celebrate its birthday, just because it’s cheesy fun for me and dogs tend to be sociable, but this cat wouldn’t be interested at all.
Yes, because they are half of a litter a feral had at a friend’s house, and they have the other two cats.
I was told that Dixie was born on my birthday.
So, No. 
March 31, 2012 (Willow, my basset hound).
Mumble, 2013 (Tank, smooth collie).
November 9, 2019 (Skye, American Mutt, from a line of American Mutts).
So, mostly yes!
Dogs: they were both purchased from a breeder, I have registration papers for them.
Goats: I know the month and year – mixed breed kids bought from a farm.
Pony: the year plus or minus two – what the vet says based on her teeth; she was a giveaway.
Horse: was foaled at an SPCA, I have most of their vet records, so I know the month and year.
Chickens: yep, they were day-olds shipped from the hatchery.
I no longer celebrate animal birthdays. I can barely remember my own. When our daughter was small we used to celebrate dog birthdays, generally with a half bagel smeared with cream cheese and a birthday candle. We called it “bageldog day”.
Maui just showed up at our house one day when he was still a kitten and just kind of adopted us. So we can guess his age to within a few months.
Nandor was born to Mrs. H’s aunt on August 18, the day before my birthday.