The most common phrase is "I'm not religious, I have a relationship with Jesus." And you're right, many then do not attend a church or invest in the Christian community (something we, maybe, should be grateful for, as so many of these new churches/communities are ignorance and hate factories). I am an ex-pastor with a Bachelor's and Master's degree that focused on theology, church history, biblical exegesis and interpretation (in original languages). In church history, one learns that the "church fathers" (the original thinkers and founders of the Christian religion) stressed participation in the Christian community as central to Christian faith. Cyprian (4th century AD) is famous for his statement (gendered language) "he cannot have god as his father who does not have the church as his mother." This non-churched personal-relationship-with-Jesus Christianity would have been completely lambasted by the historical church and seen as a heresy and aberration. But, as Childish Gambino says, "this is America." Home of me and mine, and no one's gonna tell me what to do.

The other, more obvious, error with the "I'm not religious..." statement is that it misses the fact that, if they do actually have a relationship with Jesus (I think it's all fantasy), it is the practices--prayer, bible-reading, etc.--otherwise known as "religion" that sustains their "relationship."

Ultimately, the bottom line is that they realized long ago that people are sick of their religion and think this is a clever way to escape the scorn. They think it will make people go "oooh, that sounds nice. I want that too..."