So there were no answers at /r/askscience but the relation between the two is pretty obvious. Now what could /r/askhistorians tell me more about this subject?
Yes, it is a coincidiene coincidence.
Obviously, we have no control over the number of days in a solar year: that's just a fact of the orbital mechanics of our solar system.
The division of a circle into 360° comes from the sexagesimal number system used by the ancient Sumerians. They used "60" as their base number because it's divisible by lots of factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30. This makes 60 a very useful number when doing mathematics.
And, the number 360 in the sexagesimal system has the same place as 100 in the decimal system: it's the square of the base number. So, like we modern folks use 100 in lots of circumstances (100 cents in a dollar, 100 degrees of temperature between freezing and boiling water, 100 as the base for percentages), the Sumerians used 360 in lots of circumstances, too - including dividing up the circle into 360 degrees.
It is just coincidence.
The 365.25 day year applies only to the Earth and only at this present time and only in relation to the solar year which we have chosen to use. The year of the other planets is not close to 365 days of our days or the days as seen on that planet. A Martian year is 686.98 Earth solar days or 668.5991 sols (Martian solar days) for example.
While the Gregorian calendar use the solar year as its standard other calendar systems like the lunar calendar have other cycles. 411.8 days in the lunar case. However most lunar calendar users truncate the year at 354 days to make it fit more with the 365 days solar year. The Islamic, Hebrew and Chinese calendars are lunisolar truncating a full lunar year to make it closer to a solar year.
The time period of 365.25 days also applies only now. The earth's rotation is slowing causing the day to be longer. 620 million years ago a day had only about 21.9 hours. Assuming the earth took the same amount of seconds to go around the sun (apr. 31557600) this would have meant a 400 day year. Of course over the long term the time it takes the earth to go around the sun changes also.
So time, place and culture are the things that gives a 365 day year.
360 degrees was chosen because its a very easy number to subdivide in multiple ways. There was probably some influence that it almost matched a year but ancient people knew a year was not 360 days, well before they started using 360 degrees for a circle. They chose 360 degrees for the circle because of the math. The year may have help them pick 360 over say 720 but it was not the only factor.