I ran into this issue because I noticed that this handy feature is where you can right-click VSCode on your taskbar and access pinned and recently accessed workspaces.

Then I realized that on one of my computers, this feature was not available so I googled it and found this GitHub Issue on the Microsoft/GitHub repo:

"Recent" and "Recent Workspaces" not listed in the taskbar. #63197

The issue was unfortunately closed without a solution so I thought about it and remembered that there was a Windows setting for disabling the "Recently opened items" in File Explorer.

Turns out, this feature disables the Windows "pin" feature when turned off.