Kissa is a common short form, particularly for themed cafes: jazz kissa, manga kissa, maid kissa, etc.
In general, nearly any common long word in Japan gets truncated down to three morae. Kissaten -> kissa, McDonalds -> Makudonarudo -> makudo, Brad Pitt -> Buraddo Pitto -> burapii, etc.
I currently live in Japan and I have never come across a jazz kissa, though Jazz outlets do exist in some multicultural/expat areas they aim themselves for foreign clientele. Also kissa is a rarely used term in my opinion with other slang being preferred outside 漫画喫茶。
I think jazz cafes and jazz bars are a common thing in parts of Japan, although they’re dying out. I’d say they’re decidedly western thematically, but aimed at a domestic clientele.
I believe Haruki Murakami ran a jazz bar before becoming established as a writer.
In general, nearly any common long word in Japan gets truncated down to three morae. Kissaten -> kissa, McDonalds -> Makudonarudo -> makudo, Brad Pitt -> Buraddo Pitto -> burapii, etc.