Update: Found something very interesting. I checked the previous page on the 1890 census that Enoch E. appeared in and found an Enoch "Bennum", originally of North Carolina, who was in the same regiment (2nd USCT Inf.) and company (C) that Enoch Barnett was in. He even had the same rank Enoch Barnett had when he was mustered out; that of a private. However, the enlistment dates are different. Even so, I thought for sure I'd found our man, since the surnames were so much more similar than Etheridge is to Barnett and this man, too, lived in Norfolk after the war in the same neighborhood as Enoch E. I decided to poke around on the Web some more just to make sure, though.
Turns out that the Enoch Bennum who whose rank and dates of service are listed in 1890 is almost certainly Enoch Barnham, a farmer from Camden, North Carolina who was given the rank of corporal upon his enlistment in 1863 but got himself demoted to private a year later. Turns out he enlisted at Camp Casey just four days before Enoch Barnett did the same! I found him in censuses 1870-1900. He married a woman named Emma, was later widowed, and applied for pension in 1891. This is clearly not the same person as Enoch Barnett, who according to his service record remained a private his entire career and who definitely had a wife named Josephine at some point. So, yeah, it didn't take too long to put the kibosh on the Enoch Bennam=Enoch Barnett theory.
However, I did notice that numerous men on the veterans' census on which Enoch Etheridge and Enoch Barnett appear on also have their aliases listed. There are at least four or five veterans who have the name they enlisted under next to or beneath the handle they went by in 1890. So, perhaps Enoch Etheridge's line is simply unfinished, and Enoch Barnett would have been written as his alias had it been completed?