It depends on what group of people you're talking about. If the people you're talking about all share the same language, then (A) is correct. If not, then (B) is correct, even though each child learns only one language.

If you are making a generalization about all people, you can avoid the slight awkwardness of (B) by rephrasing the entire sentence in a number of ways:

During childhood, one always learns one's own language.
During childhood, each of us learns his or her own language.
A child always learns his or her own language (while growing up).