Step-parenting upset!

Hi all,
Thank you in advance for your replies, I appreciate your opinions and thoughts.
My partner has a four (almost five) year old daughter who he sees every second weekend (he would love to her her more often however this is the arrangement the daughters mother has made). I have been with my partner for about a year and a half now and have known his daughter for almost as long. His daughter lives with her mother and her fiancé (the child's stepfather). Her mother and her fiancé also have a baby who is around 1 year old. My partners daughter started calling her stepdad "daddy" in the last 6 months or so and my partner was very upset by this and confronted the mum about it and explained that it was very upsetting and he was hurt by this. The mother explained that the child is at her and the step dads place for the majority of the time and that he cares for her and treats her as if she was his own and that the child started calling him daddy on her own accord. Despite voicing his feelings about this, the mother said that the child is free to call her stepfather whatever she wants to and she will continue to. My partner said to her "how would you feel if our daughter called (myself) mummy"? She explained that it is different and that the child would never call me that and proceeded to pretty much put our relationship down saying that it isn't as if we are engaged or have a child and that the situation is different.
I also asked my partners daughter if she has two daddies to which she replied yes. When I asked if she had two mummies she said "no"!!!! " I don't have two mummies". It upsets me that she doesn't see me as a mother figure however she sees her stepfather as "daddy". Whenever we go to pick her up from kindergarten she runs up to her dad and says I love you and gives him cuddles and kisses and she barely even says hello to me and when I say I love you on the way home in the car she ignores me. Later in the weekend she starts saying she loves me and gives me cuddles but it takes a while to get there. This happens every fortnight. I love his little girl and would do absolutely anything for her and to see her happy. I take her places and buy her things, make her breakfast, lunch and dinner just as my partner does. How do I deal with this situation? It upsets me and I feel very much on the outer. She calls her mother "mummy", her stepfather "daddy", her dad "daddy" and me by my real name.
Please help!
Thank you.