The number $360$ as the number of units into which the circle is divided has some nice properties:
it has as many divisors as a number of its size can have
it's nearly the number of days per year
But none of these properties is really "magic", so the number $360$ isn't really distinguished mathematically and may have been chosen mainly for practical and logistic reasons.
I wonder if there is evidence that the number $360$ has been seen as a somehow "magic" number by ancient mathematicians. For example, it is the number that combines the first three natural numbers $1,2,3$ with the first three prime numbers $2,3,5$:
$$360 = 2^3 \cdot 3^2 \cdot 5^1$$
Note, that the number $12$ (which happens to be the number of months per year) is the number that combines the first two natural numbers $1,2$ with the first two prime numbers $2,3$:
$$12 = 2^2 \cdot 3^1$$
And $2$ (the number of halves of the year) is just
$$2 = 2^1$$
Note, that OEIS knows the sequence $2,12,360$ under the name of Chernoff sequence but hasn't got a lot to tell about it.