Does my teenager, whose living expenses are still fully paid by me, really need a living wage?

Thank god it's none of my employer's business what my living conditions are. If I can make do with a lot less because of personal circumstances then that's good for me, but that can't be the metric to measure other people against.

part of me feels like he’s taking some of that from someone who needs it

He's taking it from his employer who certainly doesn't "need it more", otherwise they wouldn't employ your son.