Student discounts

You need to search each venue’s requirements.

For example, Uffizi Gallery says the following:

https://www.uffizi.it/en/pages/free-and-reduced-tickets

FREE ADMISSION
Proof of eligibility for free entry is to be shown when picking up your tickets.

Children under 18 years of any nationality, showing passport or id card. Children younger than 12 must be accompanied by adults

The Accadeia Gallery (David) in Florences says this:
https://www.galleriaaccademiafirenze.it/en/free-admission/

Free Admission
You can collect your free ticket directly at the ticket offices upon presentation of a document attesting one of the following conditions (ID required):

▪ Children under the age of 18 (children under the age of 12 must be accompanied by an adult);

I doubt that your college-aged young people are under the age of 18 so they don't qualify for free tickets at the Italian State Museums. Reduced entry tickets, such as those available at most State Museums (but not necessarily civic or religious institutions) for young people aged 18-24, usually apply only to citizens of the EU + some select other countries (not the US), and the information usually states that they DO check ID. For instance, this from the Uffizi:

https://www.uffizi.it/en/pages/free-and-reduced-tickets

"REDUCED-PRICE TICKET (2€)
European union citizens only, over 18 and until the day of the twenty-fifth year of age showing passport or ID. Among the non-European countries, reduced-price tickets are granted to citizens from Switzerland, Norway, Liechtenstein, Uruguay and citizens with permit cards issued for the reasons listed on the website of Italian Ministry of Foreign Affairs. "

On that same page is eligibility for free student tickets, and I doubt your students are EU scholars or are enrolled in any of the programs listed:
https://www.uffizi.it/en/pages/free-admission-teachers-students

So as Mary said above, you do need to check the rules for each museum. As far as I know, the reduced-entry rules for the Uffizi are pretty much standard at other Italian State museums. I just looked at the Accademia in Florence and yes, one must be an EU citizen for reduced tickets.
https://www.galleriaaccademiafirenze.it/en/tickets/

"European Union citizens, from 18 to 25 (until the day of the twenty-fifth birthday) showing passport or ID. The reduction also applies to citizens from Switzerland, Norway, Liechtenstein, Iceland, Uruguay and to citizens with permit cards issued for the reasons listed on the website of Italian Ministry of Foreign Affairs; € 2,00"

It does look like you can catch a break at the Vatican museums with their current student IDS.
https://www.museivaticani.va/content/museivaticani/en/organizza-visita/tariffe-e-biglietti/biglietto-ridotto-studenti.html

Does this help?

We did buy some reduced price student tickets in Venice. I had college age students and they just brought their student IDs.

Beth, it may depend on which museums you visited and how long ago. Civic and religious museums can have different ticketing parameters than State museums. They need to check the ticketing info for any they intend to visit.