Yes you omit the 'in' BUT there is a difference between talking about your birthday and the day you were born.
We use the phrase 'my birthday is on...' to talk about the day which we celebrate every year. And we don't normally say the year, so it would be 'my birthday is on the fifteenth of November' or 'my birthday is the fifteenth of November'.
BUT you have put 2004 in the above. This sounds as though somebody is trying to say when they were born. In which case we say 'I was born on the fifteenth November 2004' or 'I was born on November fifteenth 2004'.
Does this make sense?