During the setup for Bilbo's going away party, Tolkien mentions that Frodo had been in his 'tweens', a late adolescence of the twenties that last until a coming of age at 33.
Now, this reads like a comment on 'teens' to the present audience, but I thought the present conception of teenager-dom was relatively recent, or at least post-dating Tolkien himself.
Was Tolkien referencing a life stage we would recognize as 'teenage', or was he just (to his mind) making something up that happened to retroactively become a reference?
On your initial, more etymological question, he was either using "Tweens" as a shortened term for "tweenage" a word that is a combination of "between" and "teenage" typically used to refer to the early teens (websters dictionary says between 8-12, others say 10-15). It's never been a particularly "popular" word but it is in dictionaries. There are conflicting accounts though on when the word was coined, with some saying in the 40's, some saying the 80's, and still others saying that Tolkien himself was the first to use it in the 50's. The other option is that he was referring to his "Between years" ie: between childhood and adulthood.
On the question of when the modern idea of a "teenager" came to be, I would point you to the book "Teenage: The Creation of Youth Culture 1874–1945" by Jon Savage. In it, he proposes that the modern idea of a teenager was created in the mid 40's in the United States as a marketing term for companies trying to target products towards that age group. Tolkien wrote "The Lord of the Rings" throughout the late 30's and into the late 40's, so it is certainly possible, though unconfirmed, if he was aware of the term and the growing modern idea of a "teenager"