I've just noticed that moderators are required to be at least 18 years old (the very bottom of the page). How do they prove this? Do they have to send the image of a national identity document issued by their country? I thought that SE users, moderators included, are allowed to remain anonymous if they wanted (a somewhat similar discussion took place in 2014).
The 18+ rule is likely mostly to cover legal bases for SE itself, due to laws surrounding Personally Identifiable Information (PII) access. We are not required to submit any formal proof that we are 18 (like you said, that would require revealing our real identities, which is not required in any way) but it is part of the agreement that we are bound to - it's a system of trust.
To address some of the comments - kingW3, you're right. SE does make us confirm that we are 18, so if it turns out a mod isn't 18, it's not SE's fault for not making that a requirement or not making it clear. However, we don't have to show a passport or birth certificate or anything like that.
the_candyman, age actually does matter. Not only do you have to be 18 to be a moderator, but you have to be 13 to use SE. I found out the hard way how seriously they take it. My account was deleted for being underage because I offhandedly mentioned creating my SO account when I was 12... I was almost 15 at the time. They take it quite seriously. (To clarify, this was a mistake on SE's part, because they misunderstood me as saying I was still underage. The point is that it's illegal to store underage children's PII so if they don't find out and you turn 13, you're fine afterwards. I'm just saying it's taken seriously here.)
And in case you meant that age does not imply maturity, that's correct, although in addition to Xander's comment that age is, of course, very strongly correlated with maturity, it's a legal matter too. I can access some PII of users in chat and on my site and that ability can only be handed to me under certain legal restrictions (I believe largely due to the General Data Protection Regulation, a European privacy-protection policy). As someone older than 18, I am legally an adult, and thus have the legal accountability of an adult, which means age does matter. Same goes for the 13-year-old age restriction to use the site - it's to comply with the US Children's Online Privacy Protection Act.
amWhy, a lot of things here are unfortunately unenforceable, like kingW3 said - some of the moderator agreement is entirely based on trust and things like PII access, while logged by the company, is still something they can't really bind us to be unable to share because I am not a puppet of SE. This is all part of the user agreement and ToS - volunteer moderation is what keeps the site running but with that comes its risks, and it's up to each user that signs up to accept the risk of trusting people like me, a random nobody on the internet.
I'm not sure where the rest of the comments in the argument are going, so I won't bother to respond to them. All I'll say is this - we are only moderators because we were elected by people in our community who trust us, or appointed by CMs who trust us (in the case of pro-tempore moderators). The entire system itself is built on trust. If you cannot trust any of us, SE isn't the site for you.