Is it a coincidence that the Sun and the Moon look of same size from the Earth?

The conclusion that the sun and moon look the same size in the sky is most intuitive when you think about how ecipses appear to us. The relative curve shapes when the moon covers the sun, and the ways total eclipses appear to match sizes do suggest they do "look" the same size to earth observers.

Regarding seeing gigantic moons, I know what you mean. There is a weird psychological effect where objects, particularly the moon, look bigger near the horizon than at the top of the sky that is apparently in our heads and not due to any physics of, say, refraction. I've tested this for myself by pinching a large moon between my fingers and watched it visually then suddenly seem much smaller.