This is an important point. I think Peterson is a bit two-minded (in a personal struggle sort of way) about this. He seems to admire (even push) for the personal struggle for survival / and beyond that a flourishing life – the classical conservative ‘get out there and pull yourself together, man!’ tagline – much of it being ‘no-brainer’ self-help stuff. But he also feels the pull toward ‘other-centered’ concern that is the more Christian tagline and what the left would like to be seen wearing as its benevolent cloak (even if Peterson will seemingly have none of that). He also seems two-minded about systemic injustice. On the one hand (according to Peterson), white privilege doesn’t even exist. On the other hand he will rail against Islamic countries for how they treat women – which looks like systemic injustice to me! So I’m far from endorsing everything Peterson might stand for.
But regarding morality I think he endorses a kind of “Christian approach” by seeing it exactly in your terms (as socially internalized and to be prized and cultivated as such but without necessarily having to really believe in any God behind it). If I had to guess on the strength of this and a couple of other short clips on this subject, I’d say he’s essentially an atheist who wants to believe. But neither does he want to be in anyone’s box on either side of this, and his views may be presently evolving. He doesn’t trust the flimsy categories people manufacture around all these terms (and probably rightly so!) – hence the refusal to crawl into any easy label box.