Just because they can be found many places doesn't make them non-religious. The cultures that tried to do away with religion also lost common goodness, implying the two are linked.
Discussion
Empathy is the foundation to the rule of reciprocity. We can find similar reciprocation in chimps, our closest surviving evolutionary ancestors. That is non-religious, cold scientific reasoning. Engage with the scientific development in neuroscience as it relates to this if you want to have a good dialogue.
Religion has to do with the way we are made and ought to live. The fact that chimps naturally benefit from reciprocity is not evidence against reciprocity being part of religion. You may say it doesn't only exist because of religion alone, but this is not evidence against religion.
Natural law (of which reciprocity has part) is perfected by the Christian religion in going beyond a common and natural goodwill of the other and elevating to love of neighbor.
You claimed that the law of reciprocity (do no harm) “just so happens to be nearly universal in adoption, thanks to Christianity.”
Proof was provided that reciprocity predates Christianity.
In response you claim that I am arguing that reciprocity is not found in religion. I never argued that and I cited examples of ancient religions to make my original point.
If you are going to make a case that the inclusion of “love your neighbor” is some kind of newer refinement of a preexisting golden rule cross cultural standard, you can certainly argue that. Even if that is the case, you would also have to go a further step and justify why other humans are entitled to one’s love. While reciprocity is commonly held moral axiom, the command to love your neighbor as yourself is far less universally adopted.
I don’t consider these things connected. Love your neighbor seems to be more concerned with attitude. Do unto others seems more concerned with actions.
Love is by definition "to will the good of the other", and action follows if capacity exists. It does indeed involve attitude, but an attitude that does not result in according action if the capacity exists is proven disingenuous. Likewise, reciprocity is proven disingenuous if the reciprocator develops resentment.
The commandment goes above the natural law of reciprocity and perfects it.
I don't mean to imply, although I may have done so accidentally, that Christianity invented reciprocity. I do, however, claim that reciprocity isn't completely universal. Cultures in decline begin to reject it along with other elements of natural law, and so accelerate their demise.