How to completely remove the side dock appearing in activities overview on Ubuntu 20.04.1?

I have already disabled the side dock using the Extensions application and the side dock did disappear from the desktop.

However, when I open the applications launcher (using Plank dock) or activities overview, the side dock is still there.

How do I make it not appear when I launch the applications launcher?

You can use the Hide Dash X GNOME Shell extension which does exactly what you want.

Refer to the How do I install and manage GNOME Shell extensions? question if you don't know how to install GNOME Shell extensions.

The reason why the dock was still there, even though you had disabled it, is that it is a different dock. The one you disabled is Ubuntu dock, which is a Dash to dock fork, while the other is the GNOME Shell dock. When the Ubuntu dock is enabled it replaces the GNOME Shell dock, providing a dock both on the desktop and in the dash (the GNOME Shell dock is shown only in the dash by default). The Hide dash X extension hides the GNOME Shell dock.