Paul McCartney walked into the U2's dressing room and spotted a Christian Dior jacket. And asked: "Who's wearing that?". They said: "Bono". Then Paul turns to Bono and says: "Listen man, two of us can't wear the same jacket now, can we?".
Then Bono tells this story to someone else and says: He was laughing, I was too... until I realised the Beatle had run off with me stage frock. The same man who probably invented the mop top could spot a designer label by the cut of it - amazing.
1. What does it mean "had run off with me stage frock"?
2. Is he saying that Paul invented the mop top - the thing which washes the floor, or maybe it has something to do with the haircut???
Bono said Paul had 'run off with my stage frock' meaning, he had 'stolen' it.
Note that in some dialects, 'me' is used in place of 'my' as in 'me mam (mother) will have tea on the table.' In your sentence, instead of 'my stage frock', he substituted 'me'.
The way Paul wore his hair resembled the head of a mop, and so the hairstyle was called the 'mop top'.