"Character" has many definitions, some of which can be a little similar to the definition of "characteristic". To minimize confusion, I'll be selective and try to capture the essence of common usage.
When "character" is applied to a person, it typically refers to the aggregate of moral qualities by which a person is judged apart from intelligence, competence, or special talents. Another definition: "the complex of mental and ethical traits marking and often individualizing a person, group, or nation." (M-W) Key elements of these definitions are the sense that they are an aggregate of qualities, the qualities are the predominant ones that define or differentiate, and they generally refer to moral or ethical kinds of qualities. The ability to listen to people doesn't fit this definition.
In a more generic meaning, "character" refers to the dominant quality or qualities distinguishing a person or group. or the "main or essential nature especially as strongly marked and serving to distinguish." Example: "excess sewage gradually changed the character of the lake." (same source). I'd prefer not to use a sewage example in this discussion of a woman with the ability to listen to people, but I didn't write the dictionary.
This meaning is applied to a "defining" quality. If I squint and tilt my head, I could see an argument for this definition applying in the right context, but to me, it's a stretch.
A "characteristic" is: "a distinguishing trait, quality, or property." (M-W) A characteristic isn't necessarily the defining quality, just one of many that sets the person (or thing) apart. The ability to listen to people is a characteristic.
Since "character" in the generic sense refers to the essential nature of something, it might better be associated with the meaning of "is" than "has" (although that wouldn't preclude using "has" in the right context. In "She has that rare ____", "that" refers to a specific example out of multiple things. Again in the right context, this could work with "character", but this is how one would start a sentence to describe a characteristic.
So to me, "characteristic" is the much more applicable word and an improvement. I think your judgement was correct.