A hypotenuse and an acute angle define congruence?

Yes. If you know two angles of a triangle, then you know all three. Why? Because they sum to 180 deg. So you have the hypotenuse and the angles at either end then you have ASA. AAS may not be sufficient for congruence but ASA IS.

Another way of looking at it: suppose the hypotenuse is h and the known acute angle is x. Then, the side adjacent to the angle is h*cos(x) while the side opposite is h*sin(x).

So the two hypotenuses are of length h, the two sides adjacent to the known acute angle are h*cos(x) each and the sides opposite are h*sin(x). And all three pairs of angles are equal. What else do you need to show congruence?