Disable safari icon in passwords settings

Have you tried changing the default browser? Barring that, I would suggest blacklisting Safari with Jamf Restricted Software (assuming Jamf School has that function).

It's possible to change default browser in Jamf School ?

The web browser link in Passwords only allows navigation on links of that site. 

You should be able to block Safari as an app in Jamf School using either 'Safelist and blocklist', or Restrictions payload with 'Allow use of Safari' unchecked. You can also set which web browser any web clip payload should be opened with. The last depends on the app being installed of course.

I find it best to use Jamf Safe Internet for blocking specific websites on all activity of the device. Make sure not to use Safelist and Blocklist payload together with Jamf Safe Internet, as they might interfere.

I haven't found a way to set default web browser, but after installing another, you can select default web browser in the Settings - (name of browser) - Select default browser

If the web browser is for exemple, www.google.com, you can navigate without leaving any traces.

My only issue with "passwords menu" is that students browse anonymously (with no history).

In Safari, students can't delete history but with "passwords menu" they can browse anonymously.

Jamf Safe Internet would still work for this, as it works as an always on VPN on the device, working on all networks. The web traffic would be logged in the Safe Internet part. But is on-device web traffic review a thing at yours?

Otherwise, send a feedback to Apple using Feedback Assistant to make this an MDM option to restrict.