T345249: Mitigate phase-out of third-party cookies in CentralAuth explains why CentralAuth cross-project login is broken in browsers that block third-party cookies. It does not explain why it's partially broken in browsers that don't block them (notably, Chrome with default settings). We have mostly assumed that it's for the same reason, but now that we have mapped the exact level of browser support and know that it isn't, we should investigate it.
By "partially broken" I mean that when you visit a new project, central autologin works (your first pageview is logged out, but you get a notification of being logged in in the background, and your next pageview is logged in) but edge login does not (you should get logged in to all projects at the same time in the background when you log in, so when you visit another project you already have a session). The fact that the mechanism for the two is almost the same (central autologin makes a call to Special:CentralAutoLogin in a <script> tag, edge login makes a call to Special:CentralAutoLogin in an <img> tag) suggests that this might be something broken on our side rather than a browser limitation.