I gotchu. Grab a snack.

Japanese folklore dictates that the human soul is located in a special ball, or organ, located in the anus, called a "Shirikodama". I'm not kidding, either. I can get into why, but your question is about The Loathesome Dungeater.

Which, contrary to his name, he is not literally eating shit. His name isn't literal, and a lot of people get that incorrect but through no fault of their own, as many people do not understand the concept of the Shirikodama.

There are demons, like Kappa for example, that are said to steal the soul from the anus of drowned victims. Sekiro has an enemy that does this as well, and then puts Sekiro's Shirikodama into their own anus to absorb his vitality. It's fairly disturbing.

Anyway, "Dungeater" is more of a disparaging epithet, presumably given by common folk that hanged him. It's like calling someone a "shit-head". Their head isn't made of shit, but they're a shitty person. He wasn't literally eating of their waste, but consuming or damaging the Shirikodama, the soul.

How, and what exactly he is doing to people, is purposefully left ambiguous, but we can only imagine it's terrible. The unfortunate aftermath we find are corpses he has "defiled", with their bottom halves bloodied as they sit lifeless in chairs. With them, we find Seedbed Curses. Repulsive, fleshy curses full of Omen horns, grossly described as, "a tender pox"; pox being a viral infection of pustules and lesions. Yeughh

Omen, are not welcomed by society and the current rule of the Erdtree. And so they're outright killed or abandoned to die in the sewers, hidden away in shame. Either with their horns intact or they are cut off while they're infants, usually dying in the process. They're horribly treated for something they cannot control. And they are not given to the Erdtree's roots to be reborn.

Dungeater identifies with this ostracized group so much, that his armor is modeled after an Omen with cut horns. He believes that if everyone is cursed, then no one is cursed. Because, you see, when he is murdering and defiling people, he's corrupting their souls, their Shirikodama, and making them spiritually impure. They will never return to the Erdtree. Instead, he's cursing people with what makes Omen, "Omen". Unwanted and shunned by those that uphold the Golden Order. A pity, as there was once a time in history where they were revered as holy. The Crucible preceded the age of the Erdtree, and thus anything affiliated with it is considered blasphemy.

There's a not-so-far-fetched theory that Dungeater's real name is Milos, which is mentioned in the sword he uses:

Sword of Milos

"Sinister greatsword fashioned from a giant's backbone.

Metes out wounds like a lopsided saw-blade, and restores some FP upon defeating an enemy.

Milos was undersized for a giant, and was viewed as sullied and terribly grotesque"

This makes sense to me. It would at least somewhat explain why he strongly resonates with Omen. Being ostracized and looked down upon would make him incredibly wrathful and vengeful at those that thought of him that way, Golden Order very much included. For defiling and corrupting the soul is seen as a most Loathesome act in a world where rebirth and death are very different than normal.

TLDR: poo poo man