It all depends on the crew and did the ground / check-in staff make a note in the system on check-in that it is the birthday of the pax... if the ground staff did make a note in the system, it will show up in the passenger manifest, if not, then it won't, this is due to privacy issues that the passenger manifest don't usually show pax date of birth.
However, if you are a member of Marco Polo Club, I think the system automatically flag you and listed it as your birthday on the passenger manifest. So if there is a note in the system, the crew will usually do something special for the passenger, unless the crew are very grummpy that day.