I don’t think it is relevant if your father likes your mom’s boyfriend or not (as you mentioned in your first post). Your father is entitled to any opinion about her boyfriend, but it won’t change the fact that it is HER boyfriend.

Perhaps you could try to be creative or flexible and find some other ways to enjoy the birthdays of your father and yourself? Sabby had a nice suggestion, and I am sure you can find other ways to celebrate these birthdays in an enjoyable way.