My wife is Asian, but unless I think about it, she is my wife. We have been together 36 years. Am I surprised when someone points out my wife is Asian, no. It just doesn't matter.
My grandchildren are black. But unless someone says something, it simply doesn't matter to me. They are my grandchildren. When people say they don't see color, it is not that they are unaware that there are different races, what they mean is, it just doesn't matter.
When I talk to my friend and his Hispanic wife or my other friend and his Native American wife, I am not unaware of their backgrounds, I have known them long enough to know them. But that is not what I am thinking about in our conversations.
I live in southern California, where non Hispanic whites are the second largest minority. Where most people speak with an accent, including most white people. If anything there different color, accent, "look" make them more interesting.
My second youngest daughter is an unusual mix of white and Asian (half Chinese). She worked for a Chinese doctor who assumed she was Hispanic because she is fluent in Spanish, not realizing she was Chinese. I guess he never bothered to ask, it didn't matter to him, he did like the fact that he my daughter was fluent in Spanish and so could translate to Hispanic patients what he was saying.
I do gather that many people in this country life in a "black and white" world, but when you live in a world where at least 60% of the people minimally don't look like you and most of them are from somewhere else, it is not that you are blind, it just doesn't matter.
I may well be the woman in your story meant she does not use skin color to choose her friends... when she is saying she does "see" skin color, she might just mean, it doesn't matter to her. This past week we had two couples over to the house, one mixed black and Filipino couple, one Chinese couple, and my wife (Chinese) and myself (Asian)... see I do see ethnicity, but it doesn't impact who I choose to invite into my home and who my friends are.
As mentioned, I married an Asian woman, my oldest son (white) married a black woman, my oldest daughter (white) married a black man, my next daughter (white) married a white man. Can you help me pick out the pattern, because I cannot see it. My half Asian daughter dated a white man for a while, but has dated men of all races. My youngest daughter (also half Asian) number one criteria is someone she can go to church with, that really wants to be there.
If I told my son, he married a black woman, he would point out after 25 years, he has figured that out. He just couldn't see it at first... or more likely it didn't matter. I think in your example, the black woman definitely "sees", skin color has meaning to her. And it is easy to be insensitive, when some people see the world through the lens of racism and other people simply don't register skin color as more than a different shade of skin.
But we live not only in a multi racial extended family, we live in a multi cultural family (Asian, American, German, various African nations and Haiti are all parts of our extended family's culture. We love dinner when we get together and the women especially make dinners that reflect the food they grew up on. We feel sorry for people that have experienced so little of the world, when the world has so much to experience.
I guess rather than saying we don't see color, it might be more apt to say, the more different you are the more interesting you are... skin color is not the draw, ethnicity and culture. Ideally, we would love to experience the culture ourselves and seek to do so, when we can. But second best is living through someone else's experiences and getting a feel for what their lives were life somewhere else.
These days everyone defines racism differently. Traditionally, it meant meant that somehow skin color conferred certain innate abilities and qualities that made one race superior to another. I don't think most people feel that way about other people based on their skin color. I do know some elitists that feel that way about everyone, but they mostly went to Ivy league schools and were taught that, so its not their fault. I do think some people believe skin color makes them or others superiors. In our family we respond to racism when it translates into an act intended to harm someone in our family because of their skin color, whether white, black, or Asian.
But if you define racism as someone that notices their skin is a different color than others, then I guess everyone is racist except maybe blind people. I wonder if a black and white blind person went to lunch and you heard one say to the other, I don't see color, how you might respond.