In short, are Dems really capable of appreciating and/or respecting deep religious beliefs, which may contradict certain political interests they hold, as part of respecting diversity? Or can they only respect and appreciate diversity as long as it aligns with their political program(s)?
You are putting the focus on religious beliefs/believers here. This is a fairly modern American right wing line of thinking. During the anti-war and anti-nuke movements, for example, religious figures were commonly the most prominent voices. obviously, those were mainly left wing causes and it would have been no surprise at all to see the Catholic Berrigan brothers marching beside Bertrand Russell (decidedly not a believer).
Further, it is difficult to really make sense of what appears to be your underlying notion - that faith is under attack from the left. As I've noted before, there are some 400,000 congregations in the US christianity Today
By comparison, there are exactly zero Atheist Temples.
The usual manner in which your apparent protest manifests is in claims that where folks like myself might say to a particular sort of believer "Sorry, but you don't get to establish moral or ontological truths that (because of their perceived purity or because of your certainties) the rest of us must live by. Abortion, for example. None of us on the left demand that others must get abortions. We think it's your decision. We think goveernment has no legitimate role here other than health and safety (as in all medicine).
So if you consider that christians or any other faith group stand above the rest of us in righteousness and can therefore determine how we live our lives, we'll tell you to go **** yourself.
Hope this helps.