One possible reason that an installed and enabled extension might not appear on the puzzle piece drop-down is when you are in a private window but the extension is not allowed in private windows. Could you check this article: Extensions in Private Browsing.

No, that's not it. This is not a private window. That was the first thing I checked...

Edit: Private Browsing was already mentioned, I'm going to remove my answer because jscher had answered when I was composing my response. Sorry.

Just to confirm: When you are are viewing the "Manage Your Extensions" section, "Enabled" list, you can see the password manager extension there.

If you check the Customize panel, can you see the icon in an unexpected location on your toolbars? If not, you could try the Restore Defaults button at the bottom of the panel: Customize Firefox controls, buttons and toolbars.

The extension was enabled and the icon did not appear in any toolbars. The jigsaw icon used to work just fine - Firefox is not my default browser so I don't use it that much. However, when I do use it, I always need my extensions and they've always been pinned. Anyway, I ended up resetting FF and reinstalling the extension. I was hoping to avoid this as it's still not clear why this broke but at least it's working again. Very strange....