you should be able to use two Todoist accounts simultaneously in two different browsers (not two windows from the same browser, though).
Please tell me:
- Which two browsers are you trying to use?
- Have you tried clearing the cache and cookies in both browsers?
Also, please make sure that you're not signed in through the same account in both browsers and then sign out through one of them. Instead, open Todoist in both browser while logged out and then log in through each browser to each account.
Do what I do for things like this and have another broswer open. I use Firefox as my main browser, but then have Chrome running on the side if I need to login to a system with a different user.