The straight line m in the title does not mean intersection, but that the straight line m belongs to this new plane, that is to say, this plane contains the straight line m, not just intersection. You misunderstood the title.

That's it. One and only one plane is perpendicular to α.

Choose a point A on a straight line (don't find the intersection with plane α) to make a vertical line to α, and the vertical point is B, then the plane determined by B and straight line M is expected and unique.