Leafo, thank you for looking into it. I really appreciate it.
Yes, I did try to restart the interview several times in different browsers. The result was the same.
However, when I changed the country to the UK, the TIN input worked. So there's either a specific bug for Slovakia (and perhaps other countries as well), or Slovakia is disabled on purpose, but the problem is that there's no explanation why. But 2 months ago this thing worked on Steam.