Since the Background Logo extension has not been updated for the latest versions of GNOME Shell at Gnome Extension Store, you can download the extension directly from https://pagure.io/background-logo-extension/releases and install it manually.
To do this, follow these steps after downloading the files:
This guide explains how to compile and install a GNOME Shell extension that uses the Meson build system. The presence of a meson.build
file indicates that the extension needs to be built and installed into the system, rather than just being placed in your user's local extensions folder.