The taxes are for Jan 1st 2013 through Jan 1st 2014. As Bradhamlet said they ususally send the bill around August so he probably just got the bill and isn't too happy about that.
The personal property taxes billed on July 1st, payable without penalty by no later than August 31st, are for the period July 1st, 2013 through June 30th, 2014. From your post, I'm guessing you'll have owned the boat during at least 11 out of 12 months of that period hence the tax liability is, logically, yours not the Sellers although he/she may have mistakenly omitted a tax proration from your sales agreement.
Frankly, Charlotte, I don't give a dam_, but I do know that what goes around tends to come around, sometimes in spades, and considering that the cost to you, will only be a few hundred dollars--and if you can't afford that you certainly have some surprises coming as to the cost of yacht ownership--it might just be worth you paying a little something to build up a few good will chips. Or not.
As I said, Frankly Charlotte, I don't ... but I do know what a fair minded person would do, at least in my neck of the woods (which thankfully, is no longer California although my forbearers would surely be scandalized at the notion).
Here Homer Nods...