What other notion of year would people use? There's clearly a repeating cycle of that takes approximately 360 days. The seasons cycle, the stars come back to their same spots. I don't think there is any evidence that ancient people used a different version of year. In fact, given that we still have Babylonian influences like 360 degrees in our circle, I'd argue that we have continued in their tradition.

Someone on the equator might divide the orbit of the Earth around the sun into two halves, a north-sun "year" and a south-sun "year".

I have absolutely no evidence that any group did so, of course, but it wouldn't be totally absurd.

Very valid questions and I had asked the same