@glacial It’s an interesting question which will make for some fine researching and listening to music, however, @Jeruba is correct. (And The Band is a great name for a band, as is The The, eh? But that’s just my opinion).
I thin we have to fine-tune your question. You asked singer-songwriters or solo folk musicians who use a band name versus their real name? Were you thinking, perhaps, of Jethro Tull which sounds like a guy’s name but is in fact a folk-rock (ish – please don’t shoot me) band?
Or perhaps something like Yazoo, or Dollar(band) who are (were) a duo, Dollar not being their surname, as opposed to Womack and Womack, or, in a sense, Simon and Garfunkel to name but two?
Tom and Jerry. Before Paul Simon and Art Garfunkel were convinced to use their own names they went by that moniker.
This is off-the-cuff – so if you’d elaborate I’d dig a little deeper into both my musical memory and Google’s. ;-)
GQ. I’ll be returning to it.