I'm afraid that I don't assume meaning beyond the words spoken. He said nothing that is untrue.
US cultural heritage shared with Italy, let's go back to that famous Italian - Christopher Columbus and before him, the Italian namesake of America - Amerigo Vespucci. Being discovered and colonized by Italians, I'd say that's pretty significant.
Political heritage - well, you've been denying the history for the entire post in favor of your theory that Trump doesn't know history, even though he said nothing that is historically inaccurate - it just isn't worded in a way that you can accept (because you can't accept anything Trump says).
I digress, America and Italy both threw off the shackles of monarchy in favor of republic. How many countries share that accomplishment? I can name one - the ancient Romans did it, reforming the monarchy into republic with the overthrow of the king in 509BC..... Thousands of years....check.