You have a asset in project A and you want to move it to project B then normal SOP is to open project A and migrate it over to project B which is then made available under the original tree it was migrated from.
Something workable would be if working in project B and you need an asset from project A is to open an instance of the project A and drag and drop from project A’s asset folder over into project B’s asset folder then close project A.
To super size it so to speak when opening an instance a dialogue will open asking if you wish to have only the browser open as well when assets are dragged and dropped into a new folder the effect would be the same when moving instanced assets and materials from with in the host browser.
The side benefit is one could use a host project to hold all assets and then cherry pick what is needed for the working project.
You could also alter the starter content folder to host these files as well. When you create a new project include starter content (with your adjusted assets list) that way you can have it for all projects. Remove the files from starter content that you do not need if necessary.
You can find this folder in your X:/Programs/Unreal Engine/(Engine version you want to adjust)/Samples/Starter Content/Content/