Point is element of line because we sort of defined line as a set of points that have same "direction", intuitively. Or, rather, we did not define it, Euclid just let it be if I understand correctly. So, point is not a subset of a line, because point is not a set, but rather it is element of that line.
Now, is line an element of a plane or its subset?
If we define plane as a union of all lines (i., e., union of all points lying in those lines) that shares this plane-like propery we all intuitively have, then line is a subset of a plane. But if we define plane to be a set of lines then line is an element of plane. But obviously these two are equivalent. How can this be? How can something be element and subset of something?