"In my understanding, obligations that are not legally enforced, are *moral* obligations. So if we talk obligation, aren't we talking morality under a different name?"
Nietzche said that morality is virtue in need of justification, implying that virtue should not need justification. In general, in English, "morality" implies something arbitrary, often hypocritical.
A code of conduct that emerges from the realization and reality of mutual dependence is hard to call a form of morality.
"It seems to me that the foundation of social life is its material reproduction, which from the get-go required cooperative labor."
By stating it this way you are starting from an atomized picture. "Co-operation" implies people as individuals deciding to work together, but it seems to me that in a society actually organized around the material realities of their world, that sense of "individuals working together" would not apply. It seems rather that it would be a melding of the process of humand life (and consciousness) with the process of nature, and this would call for different kind of words and concepts. As you say:
"The fact that you *owed* (in the sense of causation, not in the sense of obligation) your life to others did not mean that you had to feel a conscious sense of obligation, face a conscious moral dilemma about helping others "in exchange" or else be shamed or whatever. Morality entails choice, degrees of freedom."
When people do things hand-to-mouth, the morality aspect ought to be very thin. Unless you have been socialized one way or another, and bring that along to a brink-of-collapse situation.
I don't necessarily see that pre-capitalist societies necessarily mean hand-to-mouth living. You seem to be arguing that we can only speak of morality once we have escaped the realm of necessity; but that just doesn't feel right because except in the most narrow and extraordinary conditions, necessity is not necessarily without its options and freedoms.
Joanna