Or with session ID in URL. It's a valid mechanism - works reliably, is fast, and doesn't require special support in browsers. Remember the '?PHPSESSID=...' littered all over the early internet? Yeah that one.
Of course there are some caveats, which made cookies win in the end:
* it must be applied to every internal link in the website
* it's equivalent of a "session cookie" (no persistence after browser is closed)
* it's a privacy risk when sharing the URL via copy/paste
* it's subject to 'session fixation' attack (related to the previous caveat)
Presumably the last two could be fixed by an extra HTTP header or special in-browser handling of a designated parameter, similarly to how CORS got handled.