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".

Kazinator is right. Don't think vegetarian. Think "herd animal sluggishly grazing in the fields".

"meat" = vagina