Certainly morality is a shared part of us all, it was evolved to create ingroup cohesion and resist outgroups in humans, so there are a lot of commonalities like reciprocity, justice, etc.

It’s the divvying up of ingroup and out group that tend to create disagreements in morality. I.e. we understand forgiveness, but we disagree who is worthy. We understand justice but we disagree who deserves punishment or why.

That tends to make a pretty big difference in results and conclusions.

Yes.