That is not the first time that Interrail.eu reservation service has this problems with Trenitalia reservations. For most train companies your are adult from about 14. 

If you are traveling now, go to an Trenitalia Ticket office or “Last Minute” Desk in big stations and ask for changing of the reservation into 2nd class, if there are free seats left, you will get for free the change, if you start in a station without office, ask before boarding the train staff. 

If you have time till traveling, than ask directly interrail support and I will also tag @Nanja . Maybe you could also ask directly Trenitalia via Social Media if they can help you, pleas ask them with DM on Twitter or Facebook and provide them the PNR code that you got on the reservation. 

That the reservations say you're adults is OK. The price is the same anyway.

That they say you booked in first class is strange. If you check in your Interrail account, in "My trips & travellers”, do the traveller details say 2nd or 1st class?

In any case, it's best to contact customer support. Do mention in the request when your journey is.

Finally, you can book Trenitalia day trains without any booking fees via ÖBB using these steps.

The traveller details all say second class. Even in the same e-mail in which I received all the seat tickets it said it reserved tickets in second class.

Does Interrail have a working phone number for support?

No, you will get support only via webform, if you are traveling now you will get an answer asap.

I now also see they reserved 4 seats in Business in the Italian train. But this doesn’t even fit with the first class pass, right?

Frecciarossa train in Italy have 4 classes and Frecciargento trains ETR700 (former Fyra trains in the Netherlands) have 3 classes, other FA 1st/2nd class. 

Standard = 2nd class

Premium = 2nd Plus, Interrail not accepted

Business = 1st class

Executive = luxury class, Interrail not accepted (only on Frecciarossa)

When I cancel our second class passes and book 4 first class passes, can I stil use the seat reservations from first class that I booked via my second class passes via Interrail, as mentioned above, or are the seat reservations linked to my second class pass?

Angelo, some of the tickets I received tell me the service is first class (the ones from Verona to Genova), and some others say the service is first class. So I don’t understand why they call it in two different ways themselves if it is the same.

Good questions, is the Interrail Pass Number on the Reservation?

So bussiness and first class are the same, although still called called in a different way on different routes?

One the reservation there is a PNR code, a CP code an Offer- service → Global Pass - Bussiness and it tells who the buyer was and how it has been payed.

You could send here a screenshot of the Reservation, but please without your data (Name, ...) and without the PNR code. So maybe I can check if the Pass Number is on it. 

1st class is called in almost the hole continent (except Spain, Italy and Eurostar/Thalys) and on all other italian trains except Frecciarossa and Frecciargento (only on ETR700 trains), where the 1st class is called Business.