Cannot generate app-specific password

I'm signed into appleid.apple.com. I click on app-specific passwords and the + icon to generate a new one. I put in a name and click on "Create". Then it tells me I need to sign in again with two-factor authentication. Fine. I enter my email address (Safari does this for me) and I enter my password (Safari has this stored as well — it's the same password that got me into appleid.apple.com in the first place, so I know it's correct). But I keep getting "incorrect password" — even when I methodically type it in (correctly) I get "incorrect password". All this time I am still signed into appleid.apple.com.

Can someone tell me what is happening?