Why is the angular dimension of the moon so close ...

It's not that the dimensions are the same, it's that they seem the same to us from earth.  Our view of the moon is because of the light that is reflected into it by the sun.  We cannot see the moon without the light of the sun.  The sun is very far away and doesn't seem very big when we see it in the sky, therefore when the moon (which is many light years closer) gets in the way of our view of the sun, we see an eclipse. 

It looks more like a coincidence to me. A theory about it would start like this: if A orbits B and B orbits C, would A and C appear to be of similar size when viewed from B?.  I can not see a connection between the size of the moon and its orbit. If it was just the size of a tennis ball it could evolve on the same orbit, but its angular size would be much smaller.

The moon is 250,000 miles away and has a diameter of 2,000 miles. The sun is 93,000,000 miles away and has as a diameter of 870,000. The fact that they appear to be the same size from earth has to do with the ratios of sizes and diameters and is nothing but a huge cosmic coincidence.