Looking at the documentation for text.lang, it says it should affect “all other things which are language-aware”.
However, there seems to be an open issue for formatting dates (or at least months) locally, #2840, but it looks like it’s blocked because of the Rust time
crate.
The icu-datetime package seems to add this functionality though, but it displays the full date, as opposed to just a month:
#import "@preview/icu-datetime:0.1.2": fmt-datetime, fmt-date
#fmt-date(datetime.today(), locale: "de")
// Sonntag, 13. Oktober 2