Marking something as an Asset is one part!
You need to set up the Asset library also.
Blender assets works like this.
It scans a folder on your PC, marked in the Asset library setup (Most likely your project map with all Blender projects).
In this folder, it searches for
.blend files (also in sub-maps). Open the Blender file, search for assets, and add those assets to the library.
It does not use your folder hierarchy but it uses the Asset library groups.
Adding an asset to the library, place it in a new or existing group.
This approach is also the reason you can see an asset from another
.blend and use it.
But you can not modify, relocate, etc. For this, you need to open the original
.blend files, where this asset was created.
I stored all my
.blend files on a NAS server.
But I regret this, because composing the asset list is then slow.
Opening file on the server, analyzing it. Etc.