*As a teenager playing a lot it never even crossed my mind to do it...I guess because I thought it was only for specially gifted 'musical' people or after many, many years of playing...and of course no-one did it!
But it's a completely learned skill and the great thing about it is it's completely at your level - if you're a beginner you do a simple improvisation, if you're advanced you do an advanced one. But it's hard to just 'do it' with no direction - a beginner can definitely do it but it massively helps to learn just a few simple rules first.*
I 100% agree with all of this. I didn't improvise as a child piano student. It just wasn't something you did. Now I find improvisation hard, because I am too focussed on "doing it right" and I find it hard to give the time to something when my head says it's more "productive" to practice "proper" pieces. I play guitar and piano, both of which lend themselves to improvisation, and I have enough theory (I can transpose easily) but it seems that learning to improvise at 50+ is hard.
My DS16 on the other hand is totally comfortable with improvisation. He has about 18 months of music under his belt, a natural interest in theory and he's spent many hours improvising over backing tracks (he uses a looper pedal to record chord sequences then improvises jazz and blues inspired guitar over the top). He can improvise on piano too although not as skilfully, because he's only been playing piano for 6 months (although he is already better than me which is thoroughly depressing, he can reel off Chopin nocturnes that would take me weeks to learn).
He plays in an adult jazz/blues group and is easily the best improviser - they can chuck any old chord sequence at him and off he goes. They all find it vaguely magical, because he hasn't been playing very long at all - it just seems that adolescent brains "get" this more readily. He's not an especially creative teen in any other way and prior to taking up guitar he hated music, so there's nothing you can identify as "innate". It's just long hours of improvising practice and gradually ramping up the difficulty (choice of scales and positions, speed, use of "advanced" techniques), and above all a lack of self consciousness and a willingness to make mistakes.