What exactly does this entail? I'm willing to be charitable in assuming that their use of "verify" isn't the modern usage of "give us your ID!" but I'm not enmeshed enough in the ecosystem anymore to know.

Yeah, IMO "verify" was a poor choice of wording for what this is. It has nothing to do with remote attestation or any other form of Treacherous Computing, and it has nothing to do with your real-life identity. It's just "go on your old device and confirm that the new device is really yours."

Respectfully, not even close. Verification is when I sign in from a new device, I use an existing device or second passphrase (either-or) to ensure that yes, it is me on both devices. I never have to reveal my ID, name, phone number, or email address to anyone. Not to Element, the Matrix Foundation, or the person running my home server where all my [encrypted] messages live.

My understanding is that there's two different types of verification.

Self-verification means that any new secondary devices you log into your account with will need to be verified by an existing login by way of an automatic popup that asks if you trust the device. It used to just be a Yes/No button but I think now they've added QR codes and/or emoji matching.

The other kind is verification between two different people, like when starting a direct message conversation, you might get the same emoji matching window to verify each other.