If you’ve suffered (like me!) issues when trying to login into a Solid App with different Pod service providers, there’s a way that helps you be able to let you Log In again: Clear Browser Cache & Cookies.
Additionally @jeffz stated
There is no easy path to being logged in multiple times simultaneously that I am aware of. Solid-auth-client stores authorization tokens as cookies so it’s pretty unavoidable ATM.
Also @JordanShurmer advice:
Using Incognito mode for one login might help
PS: I’ve considered interesting to track this issue into the forum (and will try to help as much as I can from now own).