It depends on the specific third party login service. Some rely on third party storage-sharing, and there are web compatibility measures built into Total Cookie Protection to allow those to keep working.

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...