This is becoming a surprisingly common question these days, especially for the Romans in particular for some reason. Here are two answers that I've written on the subject, and I believe others have commented the same elsewhere. In short, mention of birthday celebrations is abundantly attested in Latin literature. As with so much of Roman class structure, we know relatively little about the activities of people outside wealthy and educated circles, but there seems to be no reason to suppose that even the poorest Romans commemorated their birthdays with some sort of offering and prayer, though naturally the average freedman day-laborer could never be expected to afford a feast. For the wealthy extravagant birthday celebrations are attested, and at least during the Principate even those of fairly middling status appear to have been able to indulge in some sort of party--Claudia Severa, whose birthday is mentioned in the Vinolanda tablets, cannot have been above equestrian rank (and was, more importantly, apparently married to a career officer--not exactly the scum of the earth, but not real high up on the social ladder), nor was her addressee Sulpicia Lepidina, the wife of an equestrian prefect