I'm also Win 11 and drag and drop shortcut icons are correct for me - full site icons. I noticed a Windows update issue earlier regarding the same thing. You might want to check into Windows solutions like rebuilding your icon cache.

It seems this condition may not be a bug, as I see in the following link from Mozilla that shortcuts created by dragging and dropping are shown as white squares with the actual website icon as a small image within it.

https://support.mozilla.org/en-US/kb/create-desktop-shortcut-website#firefox:win11:fx104

Firefox does not control what happens on the Windows desktop. Shortcuts are a Windows creation. You may need to re-build your Windows thumbnails and icons database.