An analog signature does "prove" your identity. Not very well, of course - the amount of effort to forge it is much less than a digital signature - but that's a question of degree, not kind. They are still easier to verify than they are to forge. I think the metaphor is very useful.

It also neatly explains the concept of key sharing, which seems to present some pedagogical difficulty for some reason - you can sign your letters all you want, but if I've never seen your signature before then it doesn't actually prove anything.

Public key = knowing what your signature looks like

Private key = the muscle memory required to create a signature