I think it very much not like an organised religion - it's not organised and it's not a religion. There is no 'higher power' or 'belief'. It's all about choice, usually based on a rational assessment of available information, taken from a particular ethical position. That position varies between vegans e.g. some totally focused on animal welfare, others on the act of killing, others on environmental considerations. So it's not even a 'common set of values', though there may be a lot of overlap.
It's a response people make to the dietary choices they are faced with. (This I think is interesting, that it a restrictive response to having a lot of choice. It doesn't happen where people have very limited dietary options. Perhaps it is one way of simplifying life, when faced with overwhelming choice? But that's getting into psychology, not ethics).
I just think people get confused, linguistically, by the 'I am a ...' phrase. In their heads this connects with 'I am a Catholic', or insert other top-down belief system that one 'signs up to' by signing over an amount of personal autonomy, in favour of internalising decisions made for one by the church. 'I am a vegan' just means I don't eat certain things. It doesn't tell you why.