I’m classically trained on the violin. I have had a couple students who did this on the side. I honestly never thought you could gain much facility, neither vocally nor on the violin, doing both at the same time. My guess is the old movie you watched the character was a poor, mountain woman. It seems a novel thing since it’s not done much I think. The most recent student of mine who did that was very creative, free spirited and got a good reaction from family and friends but she never really excelled on either instrument. I personally think it’s fine for just casual fun, but in classical training, it’s not ‘a thing’ and professors of each instrument would tell you to stop that, right now, lol.
My daughter is a classically trained singer, who is pursuing a Master of Vocal Performance degree. She took lessons at a younger age and played the flute in high school. In the summers I took her to vocal workshops where she could work with professional singers. They told her to stop playing the flute because although her talent allowed her a ‘sky’ range, she sounded like a flute, they said, and she needed to bring depth into her tone and develop her lower register. So these are professional viewpoints. And from a stand point as a professional violinist, looking at my student sing and play, I understood it was a ‘pop art’ sort of thing that would effect emotional response from listeners who were untrained, who loved the sound of a violin and the sweetness of a young female voice, but this was not going to allow her to focus on the technique of each instrument. And she did eventually quit violin lessons, basically because she wanted to be self directed and do what her own inspirations led her to do. That’s what we do when we don’t have or want the wisdom of guidance.
I personally don’t think we should do anything else while playing violin, and that includes dancing, and singing, unless it is just simply a tangent and a casual, fun, random thing to do, then return to the order of practice. So I think if you keep in mind that each instrument merits separate attention, you could put them together as a novelty, but to be excellent in either, you will probably have to choose one or the other.