- 360 has 24 divisors, more than any smaller number (this is called a highly composite number).
- None of its prime factors is larger than 5 ("5-smooth").
- It can be divided by every natural number between 1 and 10, with the exception of 7. No smaller number can.
Plus a couple more characteristics that make it very much suited for subdivision. Thus, many applications of "degrees" allow the use of integers instead of fractions, which most people are far more comfortable with.
(By Remember all the people working with angles all day that do not have an academic degree to do fancy maths: Carpenters, masons, seamstresses... they prefer a natural number over multiples or fractions of some funny constant any time.
By the way, that's also the reason why we subdivide time by 24 and 60 instead of the more "natural" 10, or "fractions of day": Ease of subdivision. Thankfully, our forbearers were a lazy lot. ;-) )