I have a friend who habitually writes ‘merry birthday’ on our mutual friends’ Facebook pages (and mine, when it comes around). There’s nothing grammatically or semantically wrong with ‘merry birthday’, but it just sounds so weird. An internet search returns approx 353,000,000 results for ‘happy birthday’ and 2,650,000 for ‘merry birthday’, so it’s by no means unknown, but used less than one percent as much as ‘happy birthday’. Some of those are references to people whose birthday falls near Christmas. (I know two people born on Christmas Day. One is named Christa. I also know a father and daughter born on leap day.)
Google Ngrams shows many results for ‘happy birthday’ and ‘merry Christmas’ (of course). ‘Happy Christmas’ is used about 1/6th as much as ‘merry Christmas’ but ‘merry birthday’ yields only one result.
In the course of my research, I found this short extract (from a movie I watched more than 20 years ago, but didn’t remember this scene). The Wikipedia page for the movie says that the song was written for this movie ‘to avoid potential licensing issues’ (ie paying royalties to Warner/Chappell, at the time – for more information, see here).
(For thoughts about the song in Korean, see here.)
(PS after I posted this, the friend wished me a ‘happy birthday’ this year.)