I have the same under Windows. Tried different PCs with different OS (Windows 10 and Windows 11), tried different browsers (Edge, Firefox). Tried to reset Apple Music app, repair Apple Music app and also uninstall and reinstall, but nothing helps.
In the Apple Music app, I select plan "University Student" then click Verify Eligibility. It opens the browser https://www.myunidays.com/ which activates Apple Music window again with a smaller white window in front and nothing happens.
I use my university email for Unidays, Apple music, Unidays have verified my status.
I don't believe I do anything wrong.