I've known several violinists who played both classical and folk music on the same instrument. So I think it's all down to the musician producing the appropriate sound, not anything mechanical to do with the violin. I have to admit, it's been 20 years since I last played a bowed string instrument, and I was never very good at it. But I've been around good string players enough to watch what they do.
Left hand technique: Classical playing uses a lot more vibrato than folk. Classical violinists avoid using the open strings too much as they sound slightly different from strings stopped with fingers. Also, you can't do vibrato at all on an open string! Folk fiddlers, on the other hand, love open strings and don't mind the difference in tone- it sounds gritty and "authentic" to them. Plus there's the whole thing of using an open string to play a drone under the melody, which is very folk.
Right hand technique: You can get a lot of variation in tone with different bowing techniques. Classical players default to legato (smooth) bowing styles unless the score tells them otherwise. Folk bowing is usually less legato - there's more of an attack at the beginning of each note. I've also heard mention of folk players holding the bow in a slightly different position, but don't I know any details.
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