Ah yes, "anonymous"

To be fair, this is probably just a bad translation / bad phrasing.

I expect what they mean by this is "to continue without signing up"

It's a pretty common pattern on ecommerce sites. You can sign in with an account, or you can make a one-time purchase without an account.

And of course if you don't have an account they still need somewhere to send the confirmation (and in this case, probably send the tickets)

Step 1: Go to station ticket office

Step 2: Tell ticket agent where you want to go

Step 3: Give cash to ticket agent

Step 4: Receive ticket

Not very anonymous, they can see you.
- Wear one of those hats that cover your face in shadow so that only your eyes are visible.

Btw purchasing a transit ticket in Naples is optional, only 2 out of 5 people pay for it

Well they have to send the tickets somewhere, don't they?

Probably I should have made clear this is an app. I just expected the ticket will remain in the app.
- Ah, that makes sense then.
Lots of options that don't involve collecting data:
"Print your ticket at home." "Scan this QR code to load the ticket in an app." "Here's an e-ticket barcode you can show at the station." "Here is your PIN to access the ticket."
And so on.

They probably need some kind of token of authentication, bit In not sure what other methods they could use that would be better.

A random number that is displayed only once and it's the buyers responsibility to record it.
- Additionally, generate a downloadable pdf or similar with qr code containing whatever info needed to verify the ticket.
I can't comment on the practices of Italian transit agencies, but in the US, when you buy a train or bus ticket, the email just contains a QR code which you give to the train conductor or bus driver to scan. It also typically includes your name as well (tickets are not transferrable to prevent resale) but I don't see a reason that you couldn't design a system with simple bearer tickets.
So this system could just take your payment (which is not anonymous anyway since a name is required to process card payments) and then show you a QR code and tell you to take a screenshot of it.