One is that the login service can request access from the user using a new web API (requestStorageAccess). Another is heuristics which apply when the user interacts with the page in a way which implies a login might be taking place.
In these cases, specific third parties can be granted access to allow the login. There are some more details here: https://developer.mozilla.org/en-US/docs/Web/Privacy/Storage...
According to MDN:
> More specifically, Firefox double-keys all client-side state by the origin of the resource being loaded and by the top-level site. [1]
They linked the definition of a "site" to the HTML5 spec, which says this:
> To obtain a site, given an origin origin, run these steps: [2]
> 1. If origin is an opaque origin, then return origin.
> 2. If origin's host's registrable domain is null, then return (origin's scheme, origin's host).
> 3. Return (origin's scheme, origin's host's registrable domain).
The HTML5 spec refers to the site's registrable domain according to the URL spec:
> A host’s registrable domain is a domain formed by the most specific public suffix, along with the domain label immediately preceding it, if any. [3]
Public Suffixes are defined according to a database that you have to explicitly register in [4]. If you aren't sure whether your base domain is registered as a public suffix, then it probably isn't.
[1]: https://developer.mozilla.org/en-US/docs/Web/Privacy/State_P...
[2]: https://html.spec.whatwg.org/multipage/origin.html#site