You answered incorrectly because you ignored 子供にも. Without this 子供にも, your first interpretation would have been just fine.
Hayashi-san likes the piano, and he has been learning it ever since he was small.
However, the actual sentence does have 子供にも, and this 子供 refers to Hayashi-san's child. With 子供にも, the first choice starts to look very weird:
Hayashi-san likes the piano, and he has been learning it also from his child ever since he (Hayashi-san) was small.
(AにBを習う = to learn B from A)
This is "grammatically" correct, but semantically strange. It is impossible for a child to learn something from their own son or daughter who has not been born yet.
In your sentence, this 子供にも has a different role. That is, marking the agent (causee) of a causative verb. The following is the causative construction you needed to notice:
to make his child learn (the piano)
And the following is what the original sentence actually means:
Hayashi-san likes the piano, and he has also had his child learn it ever since she was small.