6 hours layover in Frankfurt

The United lounge passes won't admit you to the LH lounge in Frankfurt if you're sardines, i.e., economy class fliers. There is no United lounge at FRA. You could always try, but don't count on being admitted. My suggestion is that you find a decent place to get some food and coffee. Not as easy as it sounds; but with that much time, you can get to the far reaches of the airport. Maybe some posters here have recommendations. Then, find your gate and try to get comfortable in the seats there.
Your bags will be checked through to Milan, so there wouldn't be any customs check. However, as has been said many times here, customs is a matter of walking through the "nothing to declare" door, which you will do upon exiting the Milan airport.
I have changed at FRA more times than I can count, but I can't remember for sure if that's where you will go through passport control. Chalk it up to a semi-conscious state after so many hours in flight. I think it is; and, depending on how many flights come in at the same time, passport control may entail some lengthy lines. Better to have 6 hours than to sweat bullets worrying about whether you'll get through fast enough to make your connection.

Thanks Rosalyn for the clarification . Wishful thinking of me to think that I could go to LH lounge with our UA pass ( alliance ) . On a different note, is it worth it to buy the LH one day pass if they have it ?

Unfortunately no - but a UA gold

Unfortunately, there aren't any United lounges at Frankfurt Airport. However, there are Lufthansa and Air Canada lounges. I'd read the fine print on the passes and see if they're only good at United lounges or if you can use them at any of the other Star Alliance lounges.

The question of whether it's worth buying an LH lounge admission depends on how much it costs and your willingness to pay. After you see how much time you actually have, you can decide.
Btw, since those UA lounge passes have expiration dates, you might want to use them at SFO. If you allow the recommended 3 hours margin before departure, you'll likely have a good chunk of time. Those lounges have comfortable seats and a decent array of drinks and snacks.

Thanks Rosalynn, good suggestion. We will check how much time we have left and see whether it makes sense to buy the LH pass. I forgot about the expiration date on those pass . Thanks for reminding me .

With 6 hours, have you thought about going into the city and doing a bit of sightseeing, have a good German meal, and walk around in the sun? You are so close to one of the most historic cities in Germany! By train, only 11 min. and with a group day pass, it will only cost 15.80 €.