I get the joke and its self-deprecating nature. The song is originally from before the 1996 Euros as well and you had just not even qualified for the World Cup, so clearly not a hint of arrogance in its origin.
What rubs people the wrong way is the Home reference, not the winning part. Strictly correct yeah, it's probably one of the greatest positive contributions from the Empire that it spread the sport far and wide but the reactions it elicits are:
1) It isn't yours any more, sorry, it's global, it's everyone's, now go piss off with that Home shit.
2) Much like the FA chastised United for joining the EC, they actively avoided the World Cup until 1950 as if it were a lesser competition for football neanderthals/nobodies. Then they show up and get beaten at soccer by the USofA
3) I don't have an empire chip on my shoulder, but I would hazard a guess that claiming something as yours (regardless of intent, that's how it comes across) over a century later doesn't sit well with anyone who does.
Can you see how people from other cultures/nationalities could quite easily interpret it that way?
Of course 2 isn't England's fault but that of a certain type (FA authorities) which continue to piss people off to this day, foreign or English. Three is a very specific scenario, doesn't explain the Italians taking issue with it. #1 is universal though.