I'm embarrassed to say that when I was younger I always celebrated my dog's birthday. One year I put on a birthday party, inviting three friends and their dogs. I bought a can of Chum dog food (the one that comes out of the can in a solid lump,) and upturned it on a plate with a candle in it. For a present I bought the dog a jumper and a set of little vinyl boots. Yes, yes, I know - but I was young and foolish. I don't do these things anymore...well aside from a Christmas present...
Our guinea pigs were eight weeks old when we bought them, so we celebrate their birthday eight weeks before the anniversary of getting them, which is the 14th February. We don't celebrate exactly, we just acknowledge the day, and say happy birthday to them.
I think my partner and I will likely celebrate our budgies birthday's this year (their first with us as they are 1 year soon) nothing big just a new toy, extra attention and kisses and a 'Happy birthday" or two.
I haven't ever celebrated a pet's birthday. If people want to, I think that is fine. I don't think any of my pets have felt unloved because I failed to put on a party or take them to a restaurant to celebrate.