But the procedure is one thing and its meaning is another.
First let us fix attention on the purely mathematical question.

When the segment between two points A and B is congruent to that between the two points C and D, the quantitative measurements of the two segments are equal.

The equality of the numerical measures and the congruence of the two segments are not always clearly discriminated, and are lumped together under the term equality.

But the procedure of measurement presupposes congruence.

For example, a yard measure is applied successively to measure two distances between two pairs of points on the floor of a room.