Australian actor Ian Smith, when his adoptive mother discovered that she had only twelve hours to live, wrote him a letter and told him that he had been adopted as a baby. That would not have been an easy letter for her to write and it would not have been an easy letter for him to read under the circumstances, and yes, he did meet him biological mother. His biological mother, Peg Kline, thought similarly to you, when it came to him, and said, "Mum read you bedtime stories and put iodine on your knee when you fell over. My name's Peg; you just call me Peg."
I also say that there are two possibilities that should never be discounted even if they contradict each other. One is, a person's biological mother may have given them up for adoption not because they were pressured to (that's more for the older generations, though) or because they didn't want them, but because they loved their child enough to know that they couldn't provide for them in a way that somebody else could. And the other is, the circumstances of the adoptee's conception may be something they might be happier not knowing. In Ian Smith's case, he was conceived due to a rape. I remember watching SBS Insight, one night, where a young woman who was give up for adoption, was to discover, when she found her biological parents, that her father and her maternal grandfather were one and the same person.
I remember watching Losing Isiah (that brought something more complicated into the equation) and I remember having an argument with my ex-girlfriend after another situation was exposed on a soap opera whereby she believed that a child should be sent to a foster care family and returned to the biological family, and I said, "No, that's very shortsighted." She wouldn't listen to my belief, but what I wanted to tell her was, a child is not a football, and if a child develops a loving bond with an adoptive family, who is not their biological family, then it may not be in the child's best interests to be returned to the biological family.
On another side of the coin, I knew a man who was born in 1966, and was given up for adoption but how his adoption was arranged was, his adoptive mother's father was a police officer, and his adoptive mother wanted a baby and he arranged it. His adoptive father didn't know about him, because he was away with work, and he came home to find a baby in the house. His adoptive mother was a flighty sort of a person, and he described how some nights, he would have to open a tin of baked beans and heat them over a candle because he couldn't reach the stove, so he and his sister (also adopted and no relation to him) could have something to eat. His father was also something of a drinker. He did later meet his biological mother and she became a bit teary when she heard of the life he had.
As you say, family isn't always biological, and people should be respectful if a person discloses that they were adopted, and, if anything, rather than ask the person, wait until they disclose whether or not they have met their biological parents.