Why is it you translate 'munden' as his or her mouth, but this doesn't translate into 'do you like my hair?' No one in English I know would ask do you like the hair. They'd ask you like my hair.
Your five year old daughter makes you a sandwich. After a couple of bites you get something fuzzy in your mouth and your daughter asks: "Do you like the hair?".
Two bald friends are choosing a wig. The hair in one of the wigs has all the color in the rainbow and the unavoidable question arises...
I'm not a Dane, so I may be wrong on this, but I'm pretty sure that there is no implied ownership in this sentence. For example: "Hun siger at hun har smerte i ryggen" which means "She says that she has pain in her (lit. the) back" Since the subject is clearly stated, the ownership is implied. The example here, however, makes no mention of a he, she, or it for the hair to belong to. Thus "the hair" as opposed to "my hair" or "his hair" etc.