I’m pretty sure that Chrome’s alternative is designed by Google to track you in a way that’s harder to block and gives them more control over the advertising market by forcing advertisers to play along and use their method instead of collecting your data directly. Sure, it’s more private, but it’s still tracking you.
Firefox, on the other hand, is focusing on completely blocking cross-site tracking. They have no incentive to completely block 3rd party cookies as long as there is also a legitimate use case for them, but I guess they will eventually also block them if Chrome is successful in forcing websites to stop relying on them for core functionality.