Jenny wrote:
But think about it. "My woman." Not "Sarah," or "Becky," or "Jill"; not "someone I cared for" or "a woman who meant a lot to me." Very cavemannish really; it doesn't give her a name or a face or even suggest that there was anything at all between them.
No, the avoidance of her name suggests to me "too private and painful to mention".
An analogy, happily not a painful example: I once had a colleague who, when we first met, always referred to his wife as "my wife". After I had met her (and observed that they were a particularly devoted couple), he began to use her name in conversation; I concluded that when we were merely acquaintances he preferred to preserve a field of privacy around her, which he relaxed once we became friends. When Gan discusses the issue with Jenna, they're acquaintances; maybe he might have opened up more later, but at this point he wishes to give her only the basic information required.
Just listening to some Billie Holiday, which reminds me that she often sings about "my man". BH's men invariably seem to be complete jerks, but I still feel that, when she sings about them, the term conveys intimacy and tenderness.