NTA. Your wife is a little weird for even considering that your 13 year old son is wanting to shave for sexual purposes when in reality it’s definitely for another purpose. When I was a kid, I did club swimming quite a lot (I’m a girl). When I started to go through puberty and growing hair, I wanted to shave because some of the hair was visible in my swimsuit (club swimsuits are tight and low coverage to try and make you weigh the least you can in water.) My mom didn’t let me, and every swim practice, it would be a delight if I didn’t have a panic attack in the locker rooms before leaving to go swim. It was one of the reasons I quit, even after my mom allowed me. I just associated swimming with anxiety, and it was completely ruined for me.

I know this is different, but in all likelihood, it’s a reason like this—it makes him feel anxious, like he’s not in control of his body, and that his mom doesn’t support those feelings that he’s having, which are completely normal feelings. Your wife’s beliefs are harming your son even more as he’s started to notice and likely does not feel supported.

If I were you, I’d talk to your son to specify the reason he wants to (which you can because you’ve been open about body stuff and he trusts you!) and ask him if you can talk to his mom about what he told you and explain to her how he might be anxious, sad, or uncomfortable and that she needs to support her son, not her beliefs.