Thought provoking. Glad I happened upon this ( I only dip into google+ streams now and again or I'd drown in the them)
This prompted me to consider the labels that have been relevant for me (between me and the generations above and below me) - Mummy, Mum, Mother. There is also an extra label that my children used, which I associate with when they were at secondary school. They called me "Parent", and years later when it came up in conversation my daughter attributed it to the letters home which were to "Dear parent or guardian" - although on reflection now I realise that most of the time they were at secondary school I was the only parent around so it was very appropriate. I also remember my own decision when I was in secondary school to stop calling my parents "Mummy and Daddy" because it seemed babyish.
I'm called Mother and Mum by my "adult offspring". Hmm - what is the word for "children" who are no longer children? What about that expression "my children" - does the "my" have over-possesive connotations? How else can anyone signify this fact of relationship?
I relate differently to the different labels (Mummy, Mum, Mother). When the children were little I was always comfortable signing cards etc. as "Love from Mummy". Perhaps the word "Mummy" feels the same as a name. Maybe that's because I have spoken it as a name for myself, using it instead of "me" when the children were tiny. ("Give it to Mummy." "Come to Mummy" "Show Mummy" etc).
I never used the words "Mum" or "Mother" in that way and don't have the same feeling about signing those words. Those words feel like a description rather than a signature. I don't "feel like me" if I try to use either of those words in writing a card etc. I can understand why my mother changed from writing "Love from.." (when my siblings and I were little and she was "Mummy") to writing "From you loving ...." (when we were adults and she was "Mother"). What I can't remember is if she wrote it "From you loving mother" - using a lower case "m" (i.e as a description - a common noun) or "From you loving Mother" - with a capital "M" (i.e. as her name - a proper noun).
I think Morgan has chosen well. "Daddy" is what she wants to call you. It is her preferred special name for you now. It's a lovely special name for her to bestow on you. She can have permissions to call you whatever special name she chooses to call you. It might be Daddy now, and later it can be any special name she want to make up. It might even become David.
You can tell her that people sometimes have different names at different times and when she is older - in a few days, or months or years - she may want to change her choice - and give you a different name and that will be fine.
The important thing is that whatever special name she chooses to call you (out loud or in her head) she is right to know that she is very special to you. The most important thing is that you will always (to use an expression I learned from a Dutch friend) "be keeping her in a special place in your heart".
I suggest that this question about naming you "Daddy" is about one, or both, of the following.
One, she wants to be like other children she knows who have a daddy, and wants permission to claim that label for you.
Two, this is a recognition of wanting agreement about a specialness of relationship between the two of you. Such a "specialness of relationship" - especially if it is to have any long term stability (or stayability) - should be seen separately from any relationship that she has (or that you have) with anyone else (including either of her biological parents) at present or in future.
I don't really know, but that's how it seems to me.