Thanks for suggestion.
data/cache contains _dummy file only
It is not permission problem as I have ssh access and I chowned whole directory to nginx:nginx (webserver user and group).
It's not the browser cache either as I used three different browser.
It can be something in webserver itself (for example blocking some file) but in Firebug all seemed ok and I don't know Dokuwiki internals myself to see that something is not right.
In PHP error log there doesn't seem to be anything.
Because after the upgrade I saw this happening, I also tried installing a new template "monobook" which should be compatible as well (
https://www.dokuwiki.org/template:monobook) but it behaved the same - it's currently selected.
I spent quite a lot of time before writing here as I don't like bothering others. But I was not able to figure out why this happened and I am still very curious. I also tried to overwrite all files with the update zip again and I tried update plugin as well. Still the same.
Any other idea what else to try?