I saw a little message saying that there was a new version available, so I clicked it, pressed "restart now" and the app restarted. Now, for some reason, the message still pops up and even if I try to restart the app myself it doesn't even work
What platform are you on? It’s possible you have system settings or security software that is blocking the download and installation of the updated app.
You can always reinstall the latest version by redownloading the installer located here: https://itch.io/app
Just FYI, I ran into the same issue today also on Windows 10 and I think I found part of the problem.
Checking the default install location at C:\Users\USERNAME\AppData\Local\itch I found that v25.5.1 had installed correctly, but v25.4.1 had failed to uninstall.
It was this older version of the app that Windows launched when I clicked on the Itch icon or searched for Itch from the Task Bar, so every time it launched it reported that v25.5.1 was available.
After deleting this old version, I am now able to launch the most recent version as expected.