Exactly, but only because in the first prompt under the mistaken assumption that a father died in the crash, it would make sense to refer to another father. However, no father died in any car crash in your modified "riddle", which didn't stop Gemini from being anchored to the context window even after you asked it to correct itself.
Put it this way. Imagine if in the original riddle, where a father died in the accident, the surgeon had made their familial relation explicit: the surgeon could have said ‘I can’t operate on him, he is my son. I am his mother’ or, in a modern context, ‘he is my son. I am his father’. Hence, there are indeed two possibilities: the surgeon is either the boy's mother or his [other] father.
Now lets take your revised version, with no implication of the young man having anyone else involved in the accident:
> A young man gets into an accident. The surgeon, after completing his preparations, says: ‘I can’t operate on him, he is my son. I am the boy's ${PARENTAL_RELATIONSHIP}.’
Do you think that, as Gemini stated, there are still two distinct possible solutions?
Put it this way. Imagine if in the original riddle, where a father died in the accident, the surgeon had made their familial relation explicit: the surgeon could have said ‘I can’t operate on him, he is my son. I am his mother’ or, in a modern context, ‘he is my son. I am his father’. Hence, there are indeed two possibilities: the surgeon is either the boy's mother or his [other] father.
Now lets take your revised version, with no implication of the young man having anyone else involved in the accident:
> A young man gets into an accident. The surgeon, after completing his preparations, says: ‘I can’t operate on him, he is my son. I am the boy's ${PARENTAL_RELATIONSHIP}.’
Do you think that, as Gemini stated, there are still two distinct possible solutions?