Hi! Here are two threads where I explain a few more details on why Prodigy is designed to use separate processes for different tasks and why switching between different streams and interfaces in the browser like that only really works for a demo:
That said, if you're mostly interested in having a quick way to start a different task from the browser without having to go back to the command line, you could write a mini app that lets the user select a pre-defined configuration (e.g. "horses on dataset A"), spins up the Prodigy server in the background and returns the URL to the annotation app (or redirects the user).
You might also be interested in Prodigy Scale, our new upcoming project, which includes multi-user functionality, plus a whole bunch of other features. See here for details: