IE's Pinned App support (Pin to Taskbar) was better than Edge Classic's - you got a "themed" browser window dedicated to the website you'd pinned (plus tabs from there, all related to the same initial activity), and (secondary) it was usable as an Inprivate shortcut which was exclusive to that instance (i.e. you could run multiple distinct InPrivate sessions).
I'd love to see this return, so I can have my newsreader tab in its own base window, plus tabs I middle-click from articles in the news reader in the same browser window group.
The current implementation of Apps launches a new App window (good) tabless (less good) and tabs just open anywhere. Ideally, website pinned to taskbar = standalone tab group.
Thank you @TristanK for bringing this up. Being a Windows 10 Insider myself I was bringing this up for multiple times while they're were developing Edge Classic. I was one of those who requested Pin To Taskbar feature at all! I didn't get the IE style result, but still. Oh, how I enjoyed that small win of mine. :)
I would add to your suggestion @TristanK, that the (1) pinned website Taskbar shortcut should stay in a "launched" position, when you launch the pinned website from it. And (2) when you left click it again, it would find that pinned website window among all the Edge windows and bring it forward, and not launch one more website window. Only (3) middleclick should launch a new window of that website, just like you, @TristanK say. And don't forget taskbar thumbnails -- (4) all the tabs of the launched pinned taskbar shortcut should be visible. Just like IE 11 does -- this was a fantastic IE feature. And I think, (5) in ... -> More Tools, not only there has to be Pin To Desktop, but Pin To Taskbar, as well.