I have conversations about morality all the time and I am very specific about what it is and how it is we have it.
True morality comes from empathy which is a product of evolution. Too many Christians do not believe in evolution and think morality came from a book only 2000 years ago. They do not even understand what morality is, they think it deals with food, clothes and who you sleep with.
So where do you think morality comes from?
I agree that ethics are based on empathy, but they are also based on the need to protect the clan. Humans had a better chance of surviving if the tribe stuck together without internal issues, so no lying, stealing, murdering, etc. However, it could be quite acceptable to inflict those actions on/to an enemy. Getting rid of the enemy also increased the tribe's chance of survival. Some tribes cooperated and some did not.
This is abundantly clear in the supposed exodus from Egypt wherein the Hebrews killed just about everyone with whom they came into contact with.
Moses said to his followers when he came down from Mt Sinai with the ten commandments: I've got good news and some bad news guys, The bad news? It turns out is Adultery, Theft, and Perjury are off limits. The Good news? While the law says that Killing is off limits, we can ignore that one, because god said it's just a PR stunt and we can go on killing as we see fit.
Morality is, as you say, a product of empathy, in terms of the mechanism. Since different people have different levels of empathy it is mostly filtered through society so at that level it represents a sort of composite of everyone's views. In most cases the overall view of society's members is that we want society to be civil, stable, and supportive of as much individual freedom as possible, according to (roughly) the Golden Rule.
So morality "comes from" societal interactions (explicit and implicit) around how we are to coexist and cooperate. It is enabled and guided by empathy. As empathy and standards of "fairness", "equality" and other abstractions changes (hopefully, improves), societal morality also improves.
Religion appropriates societal morality and claims to be its inventor and protector but in reality it is a straight-up appropriation with a few bolted-on embellishments.
When you stop to think of it, no religion's morality can be very different from that of its host society. If it were, society would sanction it as ... immoral.
And indeed, despite claims of the religious that their allegedly superior morality is immutable, it does change. A good indicator is what fundamentalists fulminate about in any given era. Christian fundamentalists of 100 years ago railed against hemlines shorter than ankle length, listening to the radio, going to the movies, public display of affection, and un-chaperoned dating. NONE of these things is even on their radar today (except perhaps in the really extreme sects). Now it is more ideological and less behavioral ... no abortions, no reliance on social welfare (except when it's deniable), etc. But behavioral proscriptions are now mostly extreme levels of profanity and of course sexual misbehaviors (except when it's concealed).
Morality is a social construct, which is why what one society sees as Moral another society can see as immoral or amoral.
Individuals use Empathy and Ethics to choose right action, Cultures and societies adopt behavioral standards based upon these and call behaviors within that motif Moral, and ones outside that Immoral.
Thusly we see things like forced female mutilation for religious reasons in parts of the Mideast as wholly immoral, while our own society forcibly mutilates small infants for religious reasons and has zero issue with that (this is changing now).
This is reflected in a myriad of ways. We eat Beef (Moral), unless your Hindi (Immoral), same with Pork and so on, culture by culture. Each culture feeling it has it correct because people raised inside any culture tend to adopt it by osmosis when growing up in it.
Morality may have its roots in empathy but it is more: it is a set of rules that serves as an operating system for larger groups of social animals who are too intelligent to organize their lives instinctively and who are interdependent of each other.
(Book recommendation: M. Tomasello: "A natural history of human morality", by far the best account I've ever read about that topic, better than what Frans de Waal has written about it)
I really had to think hard about this when I had my children. I was Catholic; my husband was a non-practicing Jew. Both churches refused to marry us. I read my children Bible stories for the cultural literacy, but I taught them to be kind, honest, and fun to be with (non-judgmental). For me, that just about covered it.
I agree with you that morality comes biologically from evolution but it evidences itself via environment. For instance...if a child is raised in a loving home and taught the golden rule and the repercussions for actions, then that child will learn quicker than a child raised in a horribly abusive house without direction.