I run Multiple Launchers at a time, because I log into servers with multiple accounts for testing and administration.

1. Chose profile
2. Login
3. Connect to server

4. open second launcher
5. choose different account
6. connect to server

Everything here is fine. You are on the server twice with two different accounts.

7. Disconnect with first account
8. Attempt to Reconnect
9. Client responds with Bad Login.

You have to log out on the launcher with that account and log back in to reconnect.
This isn't a new issue, and the new launcher makes it easier to deal with, before you had to close the launcher but this has been going on for the last two versions.