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.
But aren't nouns linked with identity more than adjectives? If you say "I'm American" and "I'm an American", which one has more of an identity feel to it? I'd say the latter but I'm not a native speaker.