Well, I guess we must may some allowances for 15 year olds, but it seems to me that this girl has it exactly backwards. It is not "racist" to CALL your daughter
Princess, but it certainly may be racist to OBJECT to it in the way she is doing.
Princess is much more common as a given name in the African-American community in the United States than it is with other ethnic groups (though it is not exclusively African-American). Many "word names" are common in the African-American community which middle class White Americans generally don't like, including
Precious, Sincere, and
Essence as well as
Princess. This is a cultural difference. Neither liking nor disliking such names is a matter of intelligence. One's perceptions of what sort of names are appropriate or in good taste is shaped by one's ethnic background and social class, no matter who one is. It seems culturally insensitive, to say the least, for this girl to equate using a word with a positive commendatory meaning such as
Princess with something like "Dustbin Lid".