It works on 18.04, but not on 19.04.
I am not able to drag and drop Chrome or Firefox url or bookmark to the folder or desktop.
When i drag drop to the folder i get the following error: "Drag and drop is not supported. An invalid drag type was used."
When i drag it to the desktop, nothing happens at all.
Your observation is right. Drag and drop support to the desktop in Ubuntu 19.04 is limited. the reason is that the Ubuntu developers now have to rely on a very recent Gnome Shell extension to support icons on the desktop.
In version 3.30 of the Gnome desktop, which is used by Ubuntu, developers dropped the support of desktop icons from the file manager, nautilus. To continue support for icons on the desktop, Ubuntu developers shipped an older version of nautilus, 3.28, with their customized Gnome desktop. Switching now to the Gnome Shell extension to support icons on the desktop allows them to ship Ubuntu with an up to date version of the file manager.
Because the Gnome Shell extension is very recent, it still lacks features. We just will need to be patient until the extension better supports drag and drop.