"The" is used to define the entire class of instruments, therefore "I want to play the violin" is the only grammatically correct way to translate this sentence. Pointing out to a particular item (group of items) is just one use for the definite article, English is not that simple.
"I want to play violin" is not a correct sentence so it can't be translated to that. "I want to play a violin" sounds weird and nobody says that + it's not the same as "haluan soittaa viulua". "minä haluan soittaa viulua" and "i want to play the violin" mean the same thing and they are about wanting to play the violin (not a specific one, just generally). I know it's hard to translate these things when the other language doesn't have articles and it can be confusing. But yeah, I think that the "the" is needed
In English, to point at a certain violin when talking about wanting to play it, someone would more often say "I want to play this violin" or "I want to play that violin." If someone is talking about their general desire to play such an instrument, they would say "I want to play the violin" or, less formally, "I want to play violin."
It used to be in English that "I want to play the violin" was the proper and formal way, and therefore the only correct way, to state one's own desire to play such an instrument. In more recent years, it has become acceptable to omit the "the" and still mean the same thing.