Does it mean "Can you play a guitar?" is not natural? I remembered a native speaker asked me, "Can you play a guitar?'
It sounds very odd, as others have said. It would sound less odd (but still a bit dubious) if it was well known that you played the mandolin and somebody asked 'Can you play a guitar too?'. But I feel 'the' would be more natural. A better exception would be in types of guitar. 'Everyone knows you can play the electric guitar, but can you play an acoustic one?'
Apart from the case given by Ems (not talking about an ability but a position*), I would say it's traditional Br Eng rather than optional. 40 years ago I would have said 'I can play the guitar'; but today young people (exposed to the usage mentioned in my PS) often just say 'I play guitar'.
* That 'position' can be professional (in an orchestra) or functional. At the end of a rock concert the lead singer might say 'Let's hear it for/Give it up for... [='Please applaud'] so-and-so on bass... [etc.]'