"Hip hip, hooray!" is just a "cheer." It isn't specifically a birthday custom. A "cheer" is just like applause. It's a way for a group of people to express enthusiastic congratulations. It is done together, in unison, so someone needs to "lead" it. At ANY event where people are applauding, someone might say "Let's give him a cheer! Hip, hip..." and the group yells "Hooray!" In my experience in the U.S. I have NOT actually heard "Hip hip, hooray," but it could happen and it would be natural. In the U.S. the "birthday boy" or "birthday girl" makes a secret wish. She blows out the candles on the cake. She tries to do it in one breath. Supposedly, if she succeeds, she will get her wish. If it takes more than one breath, everybody politely pretends she succeeded. So, at this point, the group applauds her success in blowing out the candles. At this point people might say "Congratulations!" and "You got your wish!" and "Many happy returns!" It would be very natural for someone to say "And now, let's give her a big cheer... Hip hip... HOORAY!"