Listening back to a few tracks off of that album now...
On Fourth of July for instance, there are at least 2 tracks of the lead vocal. I can't quite pin down the panning...could be left/right/center panning with 3 vocal tracks in unison. It's very dry and very compressed with a somewhat colorful compressor. The backing instruments all provide the reverb and atmosphere..that piano has quite a bit of a very dark reverb on it.
On the title track, its very similar, with either 2 or 3 vocal tracks in unison. There could be some mid/side processing going on that's making it difficult to locate where the individual voices are panned. There may be a small amount of reverb with a short tail on the acoustic guitars and mandolin or whatever that is accompanying. But the vocals sound very dry to me until half-way through. It could be a little bit of room reverb on the background vocals, but I don't hear any reverb tail from a hall or chamber. Again here, the backing instruments like the electric piano and synth provide the reverb and atmosphere for the song.
I think that the trick to making the doubled/tripled vocals work is to get the performances as tight as possible, with every consonant and plosive and sibilant timed almost perfectly among the tracks. The minor differences of a human being performing the same part multiple times will provide the depth, but any egregious timing errors will spoil the illusion of it all being a single chorus.
When he did this album live, he had a few band mates doing backing vocals. Their voices blended very well, they're true professionals. But the lead vocal parts he mainly did by himself.