> not completely coincidentally, March was also the first month of the year in many historical calendars.

And often the last month too. The early modern English calendar began the year on March 25.

This is coincidental in relation to the offset in the names of the months. The Romans started their year in January just like we do today.

(Though in a very broad sense, it's common to begin the year with the new spring. That's the timing of Chinese new year and Persian new year. I believe I've read that the Roman shift two months backward was an administrative reform so that the consuls for the year would have time to prepare for the year's upcoming military campaigns before it was time to march off to war.)