Found the problem:

On the new server, the hosting provider automatically set the db encoding the latin1, so we had to reimport with utf8.

The WP update_options() method used in your theme seems to have problems with latin1.

There’s nothing in the console/network tabs, all ajax calls return with status code 200 and content 0. Tried with plugins disabled as well.

I am testing this on the translations page as that seems the easiest to reproduce to me. I load translation from Hungarian, all fields all filled with the Hungarian translations, I click Save Settings, success message is shown (network tab show 200 response), but if I refresh the page, the old settings are shown.