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 accounts by clicking your avatar on the upper right corner of the Fastmail screen, then selecting the other account.