>>10920956You mean apparent sizes, right?
I have a personal hypothesis that it's not a coincidence, but rather an expected result due to certain criteria for life to thrive on Earth being met.
Active plate tectonics is one such requirement. They allow for regular mixing of volatile materials in the oceans for life to begin. However, the size of the planet plays a role in this. The upper bound is a bit vague, but it could be as low as 2 Earth-masses. This is due to the fact that at that size, the gravity of the planet would significantly impact the crusts' freedom to move and thus holding back plate tectonics. If the planet is too small, then it would radiate it's heat away thus killing any tectonics before life has a change to start (there's also the loss of the magnetic field which protects the planet from solar radiation). This lower bound in mass is also vague, but Earth is clearly below it as Venus has a similar mass and it's plate tectonics are not conducive to life.
This leaves a thin range of masses of which a terrestrial planet can support life on it's own. Thus it is more likely that there will be other factors other than mass which helps with plate tectonics, such as the moon. The moon is massive enough to thoroughly heat the inside of the Earth through tidal interactions, thus replenishing Earth's internal heat as fast as it looses it. Since a relatively massive moon is required for life in these conditions it becomes much more likely that the moon would have the same apparent size as the sun. While still a coincidence, the chances are much much higher than what you described.
TL:DR The moon is roughly the same apparent size as the sun because the requirements for life make this coincidence very likely.
