I think very few people would use such a feature.
I have mentioned my religious affiliation in a few answers, comments, and on chat , where it was relevant, and I know the religious affiliation of a few other site members by similar means.
But I would be very unlikely to mention my religious affiliation in a drop-down box, or similar, for a number of reasons:
It puts me in a box, where I don't fit.
I happen to be the member of a Mennonite church. I share many of their core doctrinal beliefs and goals, such as peace and justice, a slant toward pacifism, etc. But I don't agree with everything they say or teach, and I don't want anyone to assume that I do, simply because I check-marked 'Mennonite.'
For a period of 2 years I also attended a Presbyterian church, because I was living abroad, away from my home church. Should I check-mark Mennonite, or Presbyterian? They're both protestant, but as protestants go, they likely have more differences than similarities. Should we have two check-boxes (like many sites do for spoken languages)?
I don't want people to judge me, or my answers, by my denominational affiliation.
I have answered a few questions from a Mennonite perspective. But I've answered far more from a wide variety of other perspectives. I don't want everyone to assume that, because I answer a question from the perspective of Calvinists, that my question is less meaningful, or less "correct" because I'm not a Calvinist.
Denominations, generally, focus on differences rather than similarities
This might be good or bad, depending on your view, but in the context of our site, I think it's very bad. Whether you're a Mennonite, Presbyterian, Mormon, or atheist has practically nothing to do with your ability to ask or answer questions here. We should be focusing on content of questions, not the persons writing the content. This is a core belief of the SE staff, and it's one I share for the purpose of this site.
In some contexts, it can be very meaningful to know the belief systems of others with whom you interact. This isn't one of those contexts.
Naturally the difference in denominations is important for our questions and answers, but that is where the distinction belongs.
I believe it solves a problem that doesn't, or shouldn't exist
Your stated reason for wanting this feature is "because it'll connect people with similar affiliation." This may (or may not) be true. But why is it desired?
I'll admit, when I ran across another member of our site who happened to be a Mennonite, I got a warm fuzzy feeling in my insides. But I didn't "connect" with him in any way further than saying "oh, you're a Mennonite, too?" In fact, now I don't even remember who it was (except that he was a fairly prominent member, with whom I'm sure I interact somewhat regularly, even now).
I feel much more connected to others on this site whom I know do not share my religious affiliation. If I have "friends" here, my closest friends are Calvinists, Atheists, Agnostics, and I-have-no-idea-what-belief-system-they-subscribe-to-ists. And that's perfectly alright with me.
In fact, if I were to fill out 'Mennonite' (or 'Presbyterian', or 'unaffiliated', or whatever) on my profile, and somebody sent me a message on chat saying "Hey, I saw you're a Mennonite, too!" I'd probably block them as being too superficial for me to want to communicate with them.
And lastly, the ability already exists, to the extent that it probably should
As already discussed, people can already mention their religious affiliation in their profile, and many do--especially those with official positions within their church.
This should be enough for 99% of use cases, and the other 1% I suspect fall into the categories of either "not useful" or "harmful" for reasons I've outlined above.