I really really hate this and I would put my foot down immediately. I would tell her before she comes not to bring the present. It'll keep until Christmas.
I feel so strongly about this. For most people, a birthday is the one day of the year that is 'ours'. Other days are family days, religious days or community days. A birthday is about making that person feel special and doing something for them.
Even very young children can understand that concept, and in fact I've always found that involving siblings in the whole birthday experience works really well. Mine have always loved choosing/wrapping a present for the birthday person, making a card, helping with party preparations, making cakes etc. It makes them feel just as special. Even now (aged 14, 12 and 10), they love that side of things and spend ages making cards and planning little surprises for friends and relatives, I'm really proud of them for that. And they get so excited watching someone open their presents, it's a real family ritual and they are genuinely delighted to see the birthday person get something they want.
The only tiny wobble we ever had was when our youngest was 2. He had just had Christmas and was aware of what presents were! A few weeks later it was his brother's birthday and on the big day we were having breakfast and his brother asked he could open his presents soon. Youngest then asked if he could open HIS presents. We gently explained it was his brother's birthday and the rest of us didn't have presents. His lip trembled a bit but the disappointment was over in seconds. And in fact he had a whale of a time watching his brother open his presents and playing with the paper!
I know families who are still buying presents for siblings when they are quite old. Two sisters who live by me get a 'love gift' on the other's birthday. It started when they were toddlers so they didn't feel left out, but they are now 13 and 11 and it's still the same.
Children of 3 or even younger can start to think about other people's feelings and that it is someone else's special day. Yes, there might be a bit of disappointment but by involving them in other aspects of the birthday they can get over the disappointment and enjoy the whole day.