Is a family tree considered a tree in the context of Computer Science?
No, a family tree is not a computer-scientific tree, for precisely the reason given by your professor.
A tree in computer science is a rooted (there's a distinguished node called the "root" which has no parents), directed (the edges point away from the root), connected (every node is connected to every other by a series of edges), acyclic (there is only one descending path from the root to any node) graph. The "acyclic" part implies that a node can have at most one parent, since otherwise we'd have two distinct paths from the root to the child node being considered (one passing through the father and the other through the mother).