The answers given here have opened up a secondary conundrum.
It’s uncontroversial that Turkish got the word from Bulgarian.
The controversy is whether the Slavic word came from Greek, the Greek word came from Slavic, or the similarity is a coincidence.
The Greek word could easily have come from Bulgarian; and if it’s a Slavic-wide word, that would seem likelier. There were pathways for Greek words into Russian through the Church and the prestige of the Byzantine emperors; so a word for royal garments could have made it. In fact, I was astonished to find the Byzantine clothing φουφούδιον, which I couldn’t find in any Greek dictionaries, has made it as an internet meme in Russia: Фофудья (интернет-мем) — Википедия (it’s a satire of Byzantine-flavoured hypernationalism.)
But what does the Greek evidence say?
βυσσινί for “purple” is a late inflection, it doesn’t count.
Trapp’s dictionary (LBG, Late Byzantine) has words for purple starting with βυσσιν- from the 12th century (βυσσινόχρους, βυσσινός), and a variant βυσσικός from the 9th century. The variant βυσσικός is said to refer to ὀπός, “juice”, which is suspicious. But none of these words directly refer to cherries. And βυσσικ- points away from vishne, and towards the classical derivation from byssos.
Liddell-Scott gives two definitions of βύσσινος: made of linen, already in Herodotus, and also, as the neuter βύσσινον, in the Bible (Ezra 1:6, Rev 19:8); and in Hesychius’ dictionary, meaning purple (πορφυροῦν). Hesychius preserves lots of very very ancient words. But Hesychius also has lots of more modern words—it’s an utter jumble; and my suspicion is, it’s just a recording of the Byzantine word. Hesychius was supposed to have been compiled in the 6th century, but nothing prevented later interpolations.
Kriaras’ dictionary (Early Modern) has the word βύσσινον (1638), but only in the vernacular translation of Revelations meaning “linen”, so it doesn’t really count either.
So, as far as I can tell, in Greek:
- Byssinon meaning linen is ancient
- Vyssinos meaning purple is mediaeval; possibly pre-Slavic contact, more likely post-, but still likelier to be explained by purple linen than by cherries
- Visino meaning cherry is suspiciously absent in texts older than 1600.
Hypothesis A: the shift “purple” > “cherry” happened in the Middle Ages, and was transmitted with its new meaning from Greek to Bulgarian to Russian—despite there being no evidence of the new meaning in Greek before 1600. The Balto-Slavic cognates that other respondents have mentioned should be ignored.
Hypothesis B: the meaning “cherry” came from Bulgarian vishne reasonably late, and was mapped onto the preexisting word visino, which happened to mean “purple”. (Although given Greek phonotactics, visino is the only way you could pronounce vishne in Greek anyway.) The proximity of “purple” to “cherry” is the coincidence.
Nasty when you get a battle of coincidences like that. I’m inclined to Hypothesis B. But that’s kneejerk anti-nationalist of me. Interested in others’ opinion.