I don't think it's literal; it just uses meat eating as a metaphor for aggressivity and drive. Just like in English we have "carnal act" for sex, referring to the flesh. Or whatever.

It's not even a word denoting vegetarians, but rather herbivores. A vegetarian is called 菜食主義者 (saishoku shugisha), not "soushoku" anything. "soushoku danshi" comes from "soushoku doubutsu", or "herbivore".