> Another comparison that occurs to me is joining a club.

I think it's a bit more profound than that. Club membership doesn't change most people's identity. They aren't defined by it. On the other hand citizenship is one of the first descriptors used about a person formally. Check out how any Wikipedia article about a person from the last couple centuries begins - "<Person name> is/was a <nationality or citizenship> <profession or thing they are notable for>."

"Steven Allan Spielberg is an American filmmaker." [1]

"Sir Mohamed Muktar Jama Farah (born Hussein Abdi Kahin) is a British former long-distance runner."[2]

1. https://en.wikipedia.org/wiki/Steven_Spielberg

2. https://en.wikipedia.org/wiki/Mo_Farah

I agree. It is more profound than joining a club, but conversion to a religion is primarily about beliefs. Analogies are never perfect.