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.
Really? I don't know, maybe. To me, "I'm vegan" means something like "I do veganism," whereas "I'm a vegan" means "I'm one of the vegans." Something like that.

Of course, I'm not a native English speaker, and articles (alas!) remain my English Achilles heel.
[ETA: It's even worse with French. Much worse. Sigh.] That said, I
am vegan, and that's how I normally say it (without the indefinite article).
Now, sometimes, I might say "I am
the vegan." For example, if I ordered a special (vegan) meal somewhere or other, I might say that to the waiter (so he knows who that one vegan meal is for).