The property owner's tax liability covers the calendar year, from January through December. Regardless of ownership, the tax lien attaches to the property each January and is not extinguished until the tax is paid. The first installment on that liability is due by October 1st. The second installment is due by March 1st even though this date is in the next tax year and calendar year.
The property owner's taxes fund the various jurisdictions over their fiscal year which is from July 1st to June 30th.