My husband (not the father of my child) is referred to as Daddy-Simon. He's been in my son's life from a couple of months old. It was something a relative started saying and DS picked up on it. We did correct him, but because of his age at the time we found that that actually drew more attention to it and made him do it more to get a reaction, so we just ignored it. He then started interchanging it mainly saying Simon and sometimes Daddy-Simon, but inevitably when he said it in front of my ex one day he hit the roof. I totally understood his upset and anger, but we didn't encourage it in any way and as he was only using it on occasion assumed he would grow out of it. However, after the scene my ex caused my ex then went on to come up with a mean chant for my DS correcting his use of Daddy-Simon, which my DS would come home and repeat. My response now when DS comes home chanting (it's been going on for a year) is to say 'calling him Simon is fine and calling him Daddy-Simon is ok. We are happy for you to say what you would like.' I'm not sure if that is the right thing, it feels right and it instantly settles my DS and he immediately drops it and moves on to something else.
When I discussed the chants with his dad I did explain that I understood his hurt, but that I felt the chants were wrong. I also said that as he is living with 3 other children and their mother, that I wouldn't be surprised if DS started picking up on them calling her mummy and maybe they could have a similar set up their side if that's what DS chooses to do (I cannot stand this woman for a whole long list of reasons, but my thoughts are it's a tough situation for children to deal with anyway and sometimes you've got to let them find their feet in the way that is most comfortable for them).
I firmly believe the chants have been more damaging than using the name Daddy-Simon. My son has a perfect understanding of Daddy, Mummy and Simon/Daddy-Simon. If someone says something about Simon being his Daddy he instantly corrects them and says where his Daddy lives and that Simon is his Daddy-Simon. However, the chants and anger from his dad is upsetting him. He now regularly asks me (in front of his dad) to explain what I say about Daddy-Simon, which I do.
So, in conclusion, if you've been such a big part of DSD's life, then maybe you can come up with a more familiar term for you than your name. Maybe Mama-(insert your name) or something like that? Are you in a position to discuss it with the ex? Also you could talk about it with DSD. Why are you calling me mummy? - you wont get a comprehensive answer, but you might get an insight and there's nothing wrong with involving them and asking questions, even at 3. Just my experiences and opinion, I'm not saying it's the right one...!