Translation:Matěj has neither a washer nor a dryer.
This is interesting. I (AmE) wouldn't expect "...but nor can I..." Instead, I'd expect either "...nor can I" (without "but") OR "but neither can I..." "But nor" sounds weird to me. Maybe it's a US/UK divergence? But we sort of digress from the original ani...ani issue... :-)
It does indicate that and the official translation above does use nor.
However, nor is usually used together with neither and that is what has been accepted so far. Your usage is probably less common, but looks acceptable https://english.stackexchange.com/questions/106795/using-nor-without-neither-special-case (I had too look it up, I am not a native English speaker).
On the other hand these guys seem to disagree https://www.manhattanprep.com/gmat/blog/2017/04/11/gmat-grammar-using-nor-without-neither/ I honestly do not know.