While buying and dismantling ships 15 times works, I managed to just buy a ship, leave the game when the Pirate Lord was congratulating me, and every time I re-joined my session that same ship counted as a new one.
One thing to note was that I ALREADY had a brigantine and had gone through the captained ship tutorial when I bought my sloop to do this in. But if you're doing it with your first ship try quitting the game when the pirate lord appears and see if it updates your ships owned.
So buy a ship (sloop is the cheapest at 250,000 gold).
Load into the game and run to your ship.
When the pirate lord appears and says "Ah captain, I see you're ready for your next grand adventure!" quit the game through the Xbox menu when he finishes saying "adventure".
Re-open the game, re-join your current session, run to your ship, and leave again at the same time.
Do 15x and you'll have your achievement.
I will say that the tracker didn't work 100% when I did this, sometimes it would update every time I loaded back in, sometimes every other time, sometimes after 3 times but I got the achievement after the 15.
This way you won't need millions of gold for your ship. Hope this helps.