Why don't independent singer/songwriters hold the copyright in their own name instead of some made-up company?

The vast majority of songwriters own their copyrights in their own personal names. Not sure why you would believe otherwise. Unless they have a contract with some other company, there would be no reason for their copyrights to be owned by others. On the other hand, if you're looking at the copyright on a recording of a performance, that would be jointly owned by every performer involved, unless otherwise determined BY CONTRACT. That sort of contract has specific terms about whose name is on the recording, not to mention who owns the copyright and its distribution rights, regardless of who authored the composition.

there might be a good reason everybody does it, although sometimes there isn't a good reason everybody does something, like why every girl has to get a tattoo.

could be branding. it's perceived to be easier to build a brand as a company that people might mistake for a 'record label'. than to be a solo act industry that britney spears and lady gaga seem to have monopolized, even though they havent monopolized it, because theyre actually all backed by a real record label i.e. professionals who manufactured them.