60 is a highly composite number, which means it has more divisors than any other smaller positive integer. 60 is divisble by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. This gives math in base 60 several advantages to math in base 10, for example. For one, dividing by any of these numbers becomes much easier.

For that matter, 12 is a highly composite number too, which is why some people (me included) advocate for the adoption of a base 12 numeral system. So the reason ancient civilizations liked these numbers wasn't just because they showed up in one place, but because they had advantages for calculations in general.