Why?

I think-for the IC-treni (cost 3€) it hardly matters-they wont check if it is for that pass, as long as you sit in the allocated seat/right date/right train.

For the Frecce treni (most cost 10€) it may vary-if you have some time and if you speekee/parlate the italiano enough you could ask at the stazione-after having waited a few hrs in line. 

For overnight trains its probably much more difficult

Don’t think anyone will care about the Interrail number on the reservation… as long as you have a valid reservation and an Interrail pass, there will be to 99% no problem… no one cares what number is written on the reservation.
 

A few hours in line? I don’t know in which Italian city you’ve been but I’ve never spent long time waiting in a queue… not in Firenze, Roma or Milano… 

So you can use the seat with a different pass than the one that is written on the seat reservation?

Yes, as said to 99% they won’t care about this. Main thing is having a valid reservation as otherwise you will pay an extra service fee on the train.

@Claudi. For reservations done via Interrial/Eurail for Trenitalia, the pass number is displayed on the reservation.