Multiple Logins to Office 365
Working at a managed service provider, I continuously had to login to different Office 365 subscriptions for various customers. This proved difficult and fraught with errors. My favourite error was logging in as a user under one domain, clicking the admin centre button and getting a different customer's details.
One trick is to use different browsers, i.e. Edge for one subscription, Chrome for another, Internet Explorer for another. Even this was often not enough. Another trick I found useful was going Incognito or using InPrivate tabs. This was useful to a degree. The best way though seems to be ensuring the customer has added you as a partner of record.
This can be done in two ways. The customer can do it manually or if you have admin user access to their portal, you can do it using this link:
The other way is to ensure the user is invited from the Partner Portal from the start. Follow this link for more details.
Finally, this sounds simplistic but is very effective, ensure that you untick the "keep me signed in" check box on logging in and make sure you log out of each open browser window which relates to that Office 365 session when you have finished your latest update. This sounds cumbersome but it's really not bad when you get in to the habit.