I don't have my book on the origns of math words, but I believe the root meaning is 'throw together'. To me, in geometry, it means I can take isometries and place one object on top of the other: two congruent triangles, two congruent angles etc.
I very much follow the idea that geometry is about transformations and it is valuable to not only know that there IS a transformation taking one to the other, but often valuable to KNOW which.
Consider a parallelogram - with opposite sides equal. Drawing a diagonal, you find that the two sides are 'congruent'. Which transformation? A half turn about the mid point of the diagonal! In fact ALL of the essential properties of a parallelogram follow from the fact that there is a half-turn symmetry around this mid point (where the two diagonals meet). This includes the fact that opposite sides are parallel (don't meet).
A nice thing about all this is it works perfectly well on the sphere (except, of course, that the opposite sides do meet - at antipodal points equally spaced from the 'center' of the half-turn).