No, the coat analogy doesn’t work at all because the two aren’t equivalent by any means...
If, let’s say, half of the people in the office shared a store of communal coats, all of various types and descriptions and with a range of appliqués such as first name, initial, famous characters, titles such as Best Mum or Favourite Teacher, but everyone knew that they were free for anyone to use, then one could argue that if someone chose to hang their strictly non-communal coat in the designated place of storage for communal coats, they couldn’t necessarily expect everyone to realise that that one specific coat was not for communal use, even if it was marked with a full name. Especially if the coat was well-known as a gift item that many people find quite obnoxious, become overrun with and very often do bring in to the office for the express purpose of being used by others. Then I might relate that to my argument.
But since that isn’t what happens with coats, it’s not really a valid analogy on either side of the debate.
Again, I’m going to stress I’m not saying it’s right to use named mugs unless you know the person doesn’t care, but some people will quite reasonably have a different view. We tend to presume that other people are like minded to us when that’s really not a given. You seem to be getting caught on the point that people should just know that it’s yours and for your use only, when actually office mug culture doesn’t work that way (no, not even if it has your full name printed on it, and actually, probably less so) and if you want it to be for your exclusive use you need to fall in with the way things are seen by a significant portion of the office population. That’s just how it works, and it’s pointless arguing about it here - even if everyone agreed with you it still wouldn’t make everyone in your specific office see it that way, which is where the real issue is.