I also get a "New sign-in detected for your Microsoft account" email each time with it still saying it's already linked with another account. Per Jörg's comment, I made sure my VPN was disconnected so my PS4 and computer were on the same ISP connection (just in case that was a security precaution where the IP needs to be the same), and it still gave me the same error. I've tried everything mentioned here, and I'm still getting that same error.
I tried the suggestions above (changing my password, turning off two-factor auth, removing my authenticator, logging in via non-mobile web browser, etc.) without success (maybe I did it out of sequence or I missed a detail? Seems like I followed it thoroughly enough for it to work). As such, I re-enabled two-factor and my authenticator for ideal security moving forward.
Either way, it still gave me that same error so I'm eagerly awaiting for this to be resolved properly without needing to jump through a varying list of potential hoops in order for this to work.
I, too, was able to log in after the 2.01 update (hooray!) I backed out of the remoteconnect code screen since that was taking absurdly long, but found it actually logged in (apparently, judging by my Xbox name & picture shown on the main screen as well as saying I'm signed in via Settings => Profile) & just didn't advance past that screen automatically.
Unfortunately, I, too, am currently unable to access my Profile, Minecraft Store, etc. per that still saying "You will need an account on 'PlayStation Network' to browse the store." which only offers that I log into my Microsoft account, and then it seemingly hangs at the "Signing in with your Microsoft Account..." status which I eventually cancel for it seemingly not doing anything. All of these remaining issues are wierd since I am definitely signed into my PlayStation account, and Minecraft itself shows I'm already signed into my Microsoft account (with it prompting me like I'm not signed into either when trying to access the store & just sitting there when I try to follow its prompts.)