[susan53]

Language changes. In Shakepeare's time the expression was "to play on the (instrument)". Here's a quote from one of Christopher Marlowe's plays :

(Tamburlaine the Great, Part I. ) .... "We learn," he said, "to study by studying, to play on the lute by playing, to dance by dancing, ...

Notice this is not the same as the modern use of the phrase with "on" - ie to play something on the guitar : If you're a beginning guitar player, you're probably interested in finding easy songs to play on the guitar.

So the language has changed, dropping the "on" when discussing general general ability rather than playing a specific piece of music. And I'd suggest it's changing again. It may well be that 30 years ago everyone said "play the guitar", and that many people still do. It's certainly the expression that I would use - I don't think I'd ever say "play guitar". But just because it's a part of my idiolect (and of your friend's, obviously) that doesn't mean it's "right". There is no "right" and "wrong" with this sort of expression, which is just a matter of usage. If not why do we say:
I can play the guitar but
I can drive a car

Were it a "grammatical rule", the article in each of these sentences should be the same. The definite article expresses specificity while the indefinite article expresses generality - so, logically, here we need the indefinite article. In other words, we should say I can play a guitar. But we don't - nobody does - and therefore if a language learner said it we could say it was "wrong".

The use of "the" with musical instruments is a usage rule describing what people say rather than explaining why they say it (similar eg to the fact that we say black and white rather than white and black. There's no particular logic behind it - the meaning remains the same. But we always use the former and not the latter.)

And the examples I quoted before clearly show that there are now alternative possibilities of usage with this expression. When 12,700,000 people choose play guitar as opposed to 3,400,000 saying play the guitar, how can that possibly be said to be "wrong"? At the very least it's an alternative to the older form, and by now it seems that it's the older form which is dying out.

Your friend reminds me of the mother watching her son in a military parade. "Oh look" she said, "all those soldiers are out of step except my little Johnny."