This is essentially the same as a forgotten SSMS request over here: https://feedback.azure.com/forums/908035/suggestions/32896039

It would be incredibly useful to be able right-click on a server-node in the Registered Servers list and change/set the connection for the current document window to that. You already have a custom context menu item in there ("Tab Color (Server)", so that hurdle is gone.

This idea is related to these, but not quite the same. In both cases, it seems they don't know about the Registered Server functionality, so they're basically asking for this idea, plus for you to re-implement that entire feature.
https://redgate.uservoice.com/forums/94413/suggestions/40385740
https://redgate.uservoice.com/forums/94413/suggestions/13425135