I'm starting to wonder if it's about avoiding labels (you're a... something) more than anything else. I might be wrong.
It's always seemed to me that those who say "I'm vegan" are taking the label/identity aspect to the next level, not avoiding it. Vegan literally becomes how they see themselves and relate to the world. It's subtle outwardly but has a deep meaning to them. It's who they ARE. Not what they DO.