I’m not entirely sure how I can test the difference between Mac and Windows because I can login to Facebook using my browser on either OS. The problem appears to happen only in “Internet Accounts” under SysPrefs.
Another test I did was login to Facebook using my old iPhone 5c. I entered my Facebook details inside Settings>Facebook. Low and behold, it also kept asking me to re-enter my password.
I’m positive that this has something to do with the authentication method Apple is using to connect to Facebook. Both companies should talk to each other and investigate the problem. Again, re-installing High Sierra didn’t fix my problem.