The idea behind the -cache.lib is that it's specific to a project. When you add a symbol from a symbol library to your schematic, the symbol is also copied to the project's cache library. This is how Kicad's schematics can shared without also distributing the original libraries.
Now, as you've noticed, the name of the cache library references the project. That is by design. If you wish to use the symbols in a different project, you have to add them to a standard schematic library. Using the library editor, you can load the symbol from the cache library and save it to a new library.
Yes, it's somewhat ridiculous. The developers are working on a major update to EESchema which will make it work like pcbnew. That is, when a symbol is added to a schematic page, that symbol is embedded in the page file (like pcbnew), and not referenced from a library. This won't be ready for quite some time, though.