Yes, our family does celebrate our pets' birthdays! We see our pets (dogs, cats, gerbils) as part of our family, and so we celebrate their lives with us.
For our dogs, we usually have a dog-friendly cake (no chocolate), and some gifts. They love to open presents! We do something similar for our cats--only making the snack and presents cat-friendly.
For our gerbils--I make a "cake" from snacks they like such as sunflower, pumpkin, and other seeds. I make sure to only give them very small servings and maybe some new toy for them to play with.
Celebrating our pets birthdays is a lot of fun for them and for us! And a great way to show our love for them since they are part of our family.