There's a new version of the desktop app (26.6.0) that should hopefully fix the issue. At least it did for me.
https://github.com/itchio/itch/releases/tag/v26.6.0
Viewing post in New human verification fails on app
Interesting. Had the same problem, but thought this might be something only affecting me.
I resolved this by going into "C:\Users\USERNAME\AppData\Local\itch" and starting "itch-setup.exe" there manually.
See the change date of the file, it should be the date when you tried to auto-update. So, this should already be a new installer. Start it, go through the installation process and the desktop app should launch then. Also, the installer should have deleted the previous version in the directory afterwards.
In the app, you can verify at the bottom of the settings menu that you've got the new version now. It should say: "itch @ 26.6.0".