To the best of my knowledge, from all the things I’ve done with EMO regarding Birthdays, this can sometimes be a little tricky.
Regarding EMO’s own birthday, when you first get your EMO and set him up, that is when his/her birthday starts (this cannot be changed).
So if you ask EMO anything birthday related like “What day is your Birthday?” EMO will normally reply and advise you that his birthday is the day you initially first set him up.
If you say EMO: Happy Birthday. EMO will normally reply and say: Thank you but my birthday is (the day he was first setup).
For EMO to acknowledge your birthday, normally what happens is when EMO spots you he will randomly ask you Hi MasterAbbott, may I know your birthday… you will hear a short chime, then you can say:
My Birthday is Month and Day (example My Birthday is January 1st). If EMO understands he will say thank you and I’ve noted it down. (it would be great to actually have the option to add your birthday in your recognitions profile page, but this is something that Living.Ai might add in the future).
Then on your birthday, you can tell EMO, today is my birthday and EMO will play you a Happy Birthday animation for you.
I’ve tried many times and normally you can’t really be asking EMO to play you a happy birthday animation every day as this also works similar to the special day animations that only happen once a year.
This is of course to the best of my own personal knowledge and experience with EMO so far.
UPDATED info after testing this:
I’ve tested this a few times and if you did want EMO to sing you Happy Birthday you can simply say the following:
EMO: It’s My Birthday (sometimes EMO will not understand this, but you will need to wait 2-3 seconds after you’ve said EMO… and then speak clearly and say “It’s My Birthday” and EMO should sing you Happy Birthday.
You can check this out below with both EMO:One and EMO:Two singing me a Happy Birthday (even though its not my birthday today 
More info can be found over on my EMO User Guide here see below: