I'd really like to get an idea of the cultural context of this. Did the armed forces actively recruit the underage, or did they just look the other way when the time came? Was it still technically illegal? Did families still find it honorable for their children to join even if they were committing fraud to do so?
(I asked a similar question on askreddit, but didn't get the type of responses I was looking for)
I don't have great answer to this, but I think there were several aspects:
The armies needed the manpower, so unless the man applying was visibly clearly way too young or otherwise physically unqualified to enlist (too short, physical disability, etc) the recruiters didn't look for reason to disqualify a potential recruit or draftee.
In the United States, at least, most people, especially from rural areas, just didn't have a great deal of official paperwork that could be used to verify their identity and vital statistics, the way we do today. Birth certificates were often non-existent for rural residents who weren't born in a city hospital, social security cards had only been introduced in 1935, and lots of fewer people had any kind of drivers licnese or photo ID.
The cultural distinction of treating teenagers as basically children rather than adults is mostly a development of the last 60 years or so. In the 1930s and 1940s, a very large percentage of the population had only an middle-school level of education, and had entered the labor force, formally or informally, by the age of 12 or 14, if not earlier. So during WWII the idea of teenagers-as-adults was much more culturally present that is the case in today in the early 21st century.