Whatever shape it takes on, it's a Google-only thing. Being a Google-only thing means that it will (maybe) solve the password problem... for Google.

So which desktop is runs on, or even phone, tablet or any other form factor, is irrelevant.

From the article, it seems google-agnostic:

So they’ve developed a (as yet unnamed) protocol for device-based authentication that they say is independent of Google, requires no special software to work — aside from a web browser that supports the login standard — and which prevents web sites from using this technology to track users.

So... change "Google" to "web browser". That is still a scope constraint.

I guess the main motivation for this is Chrome OS.