Unless the situation calls for it - I don't bring up my religious beliefs in conversation, ever. Especially not with strangers. I do, however, have a very visible tattoo (when I am wearing short sleeves, anyways) that would probably make it pretty obvious to others that I am a Christian. But the tattoo is for me, and is significant for me. It's not for others, or to advertise my faith.

And yes, I do think bringing it up in every conversation like that would push people further away. In fact, it even makes me uncomfortable at times, and I am a Christian. But for some people, including some of my family members, it's so engrained into their personal identity that it'd be unnatural for them to not talk about it regularly/routinely. Some people might do it as a way to proselytize/evangelize, although I think it's a very bad tactic to do that with.

But it's really hard to say, to know what someone's intentions and motivations and reasons are. I just try to see the best in people and what their intentions are, and remind myself that patience is something we all can benefit from, and that we all often need to work on.