I think you are probably right and I would throw the mug in the garbage. Not your garbage - the big dumpster outside!
You are probably having a hard time letting it (the feelings) go because it is sitting in front of you every day staring back at you.
Yes it's just an inanimate object, but it's a reminder. As for it being a gift. That's something to work on there I think? It's a coffee mug, not her Father's graduation ring (or some other family heirloom).
And it's ugly lol