Having multiple terminal sessions via serial connection is not possible. You can only have one session open. The terminal communication is through a serial port, once you have opened it for the first terminal session, it cannot be "re-opened" for an additional session because it doesn't use another serial port, it uses the same one. The data would be simply replicated in each session, as you described.
On the other hand, when you use SSH, you are using the IP address, not the serial port. In this case you can have multiple sessions; all connected to the Edison's IP address.
How about to use GNU screen or tmux?
You can run multiple shell session on Edison by using screen/tmux ON your Edison. (not on your workstation)