Electric field 'lines' were invented by people, to have a visualization of the
electric field. We draw 'lines' to show the direction of the force that a tiny
test charge would feel if it were at any point in the field, and the direction
it would move from there.
If 'lines' crossed, then there would be a point in the field where a tiny test
charge would feel a force in two different directions, and it would have a
choice of which way to go from there. That doesn't happen.