The Fastmail web interface and mobile app let you log in to multiple accounts at the same time.
When you go to log in to Fastmail, you will be brought to our Select user page, which shows any accounts you are currently logged in to. Select one to go to the account's Inbox.
To go to another account that is not currently logged in, click Log in as another user at the bottom of the list.
You can quickly switch between your logged in users by opening your settings dropdown menu, then clicking Switch user. This will bring you back to your login screen, where you can select a different account without logging out.