In a nutshell, the taxes are assessed based upon ownership as of October 1st, which is why it shows 2018. If you acquired the vehicle before October 1st of 2019, but after October 1st of 2018, then you will be responsible for a portion of the tax (a "pro-rated supplemental motor vehicle tax") based upon the amount of time you owned the car from prior to the next Grand List in Oct. of 2019.
This is laid out on page 7 of the document that adjusterjack referred you to (under "Billing and collection").
EDIT: You will ALSO be responsible for tax based on the 2019 Grand List - but that will be the full amount, not a prorated amount.