I may be wrong but I think celebrating birthdays is something that was introduced into Irish culture.
In the old days, at least in rural Ireland, people didn’t celebrate birthdays. Some people didn’t even know their birthdays.
It was the baptism day that was important. And the feast day of your saint.
The birthday didn’t become important until one had to have it for official reasons like applying for the pension.
So when we started celebrating birthdays we just used “happy birthday” 🙂