He's a mutt and he's black. The two are in no way mutually exclusive. He is of mixed ancestry, therefore he is a mutt, but some of that ancestry is African and he has comparatively dark skin and he identifies as black, therefore he is black. And in so far as this is a joke, it is not a joke in the "lol I'm not really a mutt guys, I'm totally black like the life-giving soil" sense; it's a joke because it's a play on words, since mutt in the sense of dogs and mutt in the sense of humans are somewhat different concepts, and therefore this is a humorous thing to say. If people were uncomfortable (and they did seem to awkwardly look at their shoes as you note) it seems quite likely that it was because people are uncomfortable about race in general getting mentioned out of nowhere, not because they are uncomfortable about that particular racial designation.
Sorry but you are misunderstanding what I said.
I am not saying they SEEMED uncomfortable about the designation per se. There are many reasons why a person can be uncomfortable after hearing that designation. Let me enumerate them.
1. It seems unseemly for a president elect to denigrate himself that way.
2. It might have been a term frequently used in a derogatory manner by some present.
3. There might be people their of mixed ancestry who don't appreciate that term.
4. The white folks there want to see him as totally black and their status quo sensibilities were jarred.
5. There might have been black people there who want to see him as totally black whose sensibilities might have been really jarred.
6. People there could feel a sense of having arbitrarily imposed a label.
7. Some there might be guilty of using the one drop rule to persecute minorities and were reminded of it by that word.
8. Some present might have been objects of the persecution based on the one-drop rule.
All these together might have occurred simultaneously. No?
As for the term Mutt, when used by humans toward human means multi or biracial heritage. Look it up in the dictionary and see for yourself. Then again your dictionary might differ from the usual one. Which is understandable.
BTW
The African American community didn't accept Obama as wholeheartedly theirs initially. Previous to his being perceived as having a good chance to be president, the black community was debating whether Obama was black enough or not given his mixed race ancestry. They were also debating whether he was American enough given his African father and his growing up in the boondocks of the American states.
It was only AFTER they got a whiff that he really stood a chance that they decided to change their song and tried to appropriate him to themselves. They also did the same act politically many leaving the Clinton side only when absolutely sure Obama stood a chance after making sure there were enough whites supporting him. So much for racial unity and loyalty!