As to the original post, it seems odd, very odd indeed. I certainly wouldn't take a child to a party with a gift for them and I can't imagine making the statement, (as the OP's friend did) that while your children have to have birthdays for something special, hers gets one just because. And wrapped to boot! Weird!
Siblings getting gifts on birthdays I find very tricky. I have ten year old twins and a nine year old, all with birthdays in April. Because they're so close, we do tend to get a little gift for those not having the birthday. It's usually a book, but it's never something that costs more than pocket money. And I'll admit, we do this in my house because it's not just one person's special day but two at the beginning of April, with number 3 being left out. And because we've done it for 3, we do it for 1 & 2 at the end of April.
Writing it down seems strange. I keep trying to work out how I justify it and I'm not sure that I actually can. It just feels right to me to add in a little something for the one left out, (and it is always little) when there are two who are 'special' that day. I'm certain that it wouldn't strike me to do the same thing if I only had two - my sister and I are six years and three days apart and we certainly didn't get gifts on the other one's birthday.
Oh well, I can't justify it, but it feels right to me to do it. Maybe I'll have to reconsider it this year now they're getting older and won't feel left out the same. They always seem so pleased with other people getting gifts and in making their own - I must be getting something right!