That’s interesting, because “the apple doesn’t/didn’t fall far from the tree” is a known Anglophonic saying that basically means that a child turned out a lot like a parent (gender not necessarily specified). I wonder if one is a calque of the other.
The above poster isnt really correct. We have an actual saying that is the literal translation: "Der Apfel fällt nicht weit vom Stamm ". And it means exactly what you suggest, a child being very much like one of their parents in one way or another.
Like father, like son exists as well, “Wie der Vater so der Sohn”.