What is the Islamic significance of the color green on the dome of the Prophet's tomb and in the Saudi flag, and does it symbolize something specific or is it a coincidence?

There is no specific virtue associated with the color green, but some scholars have recommended wearing it. As for the green dome over the Prophet's grave (peace and blessings be upon him), it is a later innovation and did not exist during the time of the Prophet or his Companions. Perhaps those who made it green did so out of an aesthetic preference for the color, or based on evidence recommending the color green.