With, names, he and she, the word used is 'has'. So, he has, she has as well as Mary has, Anne has is the normal usage in English. So, a question by putting does in the the front in my opinion should not change this usage of has.
Are you saying "Does Anne has an umbrella ?" should be accepted ? It isn't a grammatically correct sentence, as only one verb should be conjugated:
he/she/Anne has -> does he/she/Anne have ?
"to do" is conjugated, not "to have", which remains in the infinitive.
I forgot the question mark. Normally that is acceptable to the little owl, but not on this occasion