Common sense tells us a year is conveniently divided exactly into four seasons. I'm wondering whether that means the earth's tilting cycle is equal to its orbital cycle. If so, there must be a reason. Do we have a proof?
There is no tilting cycle.
The Earth continues to tilt the same way all year round. The axis points towards the pole star. But as the Earth goes round the sun, in December the northern hemisphere points away from the sun. In June the Northern hemisphere points towards the sun, as the Ascii art below shows (not to scale)
N N \ \ O * O \ \ S S Earth Earth in Dec Sun in Jun
So no special explanation is required. The Earth points the same way all year, it just moves relative to the sun.
The tilt of the Earth relative to the stars doesn't change. So the tilt of the Earth relative to the sun is entirely due to the Earth's orbit. This proves that the cycle of tilting relative to the sun will take exactly one orbit to complete. It is no coincidence; the cycle of seasons will take exactly one year.
(There is variation in the direction of tilt, called precession, but this is a much slower cycle, lasting 25000 years)
It is no coincidence that the Earth's 4 seasons fit exactly into one orbit of the sun. As everyone knows, the Earth rotates on its axis every 24 hours, and this turns it into a gigantic gyroscope, so despite a very slight wobble, the north pole of its axis always points more or less at the star Polaris. This means that in the summer the northern hemisphere is tilted towards the sun, so the sun is higher in the sky. At the equinox, both northern and southern hemispheres get equal daylength and equal amounts of solar radiation. Then, because the north pole of this gigantic gyroscope is still pointing at Polaris, the southern hemisphere becomes tilted toward the sun and the northern tilted away from the sun, so we have our winter while the southern hemisphere has its summer. You can work this out for yourself by using a table lamp to represent the sun and a tennis ball marked with a north pole and south pole to represent the axis. You will see that as you walk the tennis ball around the lamp, taking care to keep the tilted axis always pointed toward an imaginary Polaris, first one hemisphere and then the other will be tilted toward the lamp. The seasons have no option but to fit precisely into one orbit.
I'm wondering whether that means the earth's tilting cycle is equal to its orbital cycle.
The Earth's "tilting cycle" is about 26000 times longer than its orbital cycle, which is one year long. This means there is very little change in the seasons from one year to the next, or even one century to the next. Even after 13000 years (half of a cycle), the changes are subtle. January will still be winter in the Northern Hemisphere, summer in the Southern Hemisphere, and July will still be summer in the Northern Hemisphere, winter in the Southern Hemisphere.
What will change in 13000 years are the durations of the seasons. Currently, the Earth is closest to the Sun a couple of weeks after the December solstice and furthest from the Sun a couple of weeks after the June solstice. In 13000 years the situation will be reversed, with Earth aphelion occurring near the December solstice and Earth perihelion occurring near the June solstice. This will make Northern Hemisphere winters longer and colder than they are now, summers shorter and hotter.
This 26000 year cycle is one of the key reasons the Earth currently undergoes long periods of glaciations ("ice ages") interspersed by (typically) short interglacials. (The current interglacial is atypical.) A glaciation starts when snow in high northern latitudes doesn't melt over the course of a long but cool summer and later ends when all of that accumulated snow and ice finally melt more than they do accumulate over the course of a short but hot summer.
That depends on the definition of a year. Compared to the annual cycle of the Sun's declination north and south of the equator:
The US Naval Observatory article The Seasons and the Earth's Orbit explains tropical and anomalistic years further. The 21,000 year cycle mentioned there is a combination of axial precession (26,000 year cycle) and apsidal precession (110,000 year cycle).