It means that whoever owns the ship, and it could be the captain or it could be a merchant or a consortium of merchants or a nobleman (although that’s less likely), is based out of Oldtown. 

In terms of taxes and fees, you’re dealing with two things. The first is that the owner/s pays taxes to House Hightower by virtue of being a resident of Oldtown, but that’s no different from any baker or blacksmith who also is a resident of Oldtown. The second is that when the ship is in Oldtown, it pays tariffs and excise taxes, harbor fees and warehousing fees, the same as any other ship in the port (although it’s quite likely that the Hightowers charge a lower rate to resident merchants as a way of stimulating local commerce). 

Looking at the sources, it doesn’t seem like the Hightowers have a proprietary trading fleet in the same way that the Redwynes do – rather, the Hightowers’ navy is meant to “to protect his trade.” Nor does King’s Landing have a proprietary trading fleet; they have a lot of ships based out of that port, but the crown doesn’t own them directly, although Littlefinger probably bought quite a few of them using embezzled royal funds.