Call them both daddy.
Joe (Or whatever his name is.) is also your daddy.
This should be all the explanation your two-year-old needs. She is not likely mature enough to understand the complexity, and won’t be for a long time. So a simple explanation, “some girls get two daddy’s,” is enough for now.
This is a perfectly normal practice for young children with step-fathers and bio-dads. Even for older children who understand there is a bio-dad and a step-dad.
It sounds like the biological father is trying to reconcile. Put your personal feelings aside and focus on building relationships based on honesty — which is surely the best example you can set for your daughter. If the biological father is actually supporting his daughter, there is no reasonable explanation for denying him this ultimately small request. It is, after all, just a label; however important it may seem.
He will always be her biological father, never telling her will not change it, and it sounds like he also wants to take responsibility and support her.
There is the possibility that she will greatly resent you for not telling her important information about herself, if that’s what you are suggesting, when she is older.
That is not the case for your boyfriend. And, I don’t be to be rude or inconsiderate, my next statement may never be tested, and I hope it isn’t, but there is no guarantee the relationship will last; unless he legally adopts your daughter, there may be a point where he is not in the picture.
Better to tell her of her reality while she is young and able to adjust easily, not when she is 18 and it falls like a bombshell.