I did a quick check in the dictionary and on a grammar forum and teenage and teenaged convey the same meaning and both are correct. The only point to remember is that we generally use this adjective in front of the noun "He is my teenaged son." not "My son is teenaged." Where I grew up (Ohio, USA) people said teenaged, but it seems like teenage is more common in some other places (apparently England is one). I usually write teen-aged. This was historically correct (I found a bunch of references writing it this way), but that might have changed. You are probably best to just write teenage/teenaged. "He was in his teenage years." and "He was in his teens。“ are again both correct and convey the same meaning. "He was in his teen years." might be correct, I am not sure, but it sounds like something I would read in a novel from the 1940's or 50's. I wouldn't use it in daily conversation if I were you.

The confusion lies in speech patterns being shortened....thusly becoming less formal also. You may say " in his teens" as opposed to " in his teenage years" or " during his teens" " during his teenage years"