Pirates of the Caribbean (2003) is my favourite pirate game. No, it's nothing like what you think a licensed tie in game from 2003 would be like.
It's a real oddity. This was made by a Russian studio and originally meant to be a sequel to their previous age of sail game, Sea Dogs. In Russia it was still marketed as a sequel, without the Pirates of the Carribean branding. It has basically nothing to do with the movies in reality. I have no idea how or why this ended up being a tie in.
I don't really have a short hand for describing the genre. It most reminds me of space sims - where you control a vessel which you can replace, has an economy/trade system, management mechanics, factional reputation systems and an open world. It's not a simplified as Freelancer, but not a spreadsheet game like the X series.
The sailing is great, a happy medium between completely arcade stuff where you just point your ship where you want to go and sims. Wind and weather play an important role without being tedious or overwhelming.
You also control a character for ship environments, like boarding, and exploring towns and islands (with swashbuckling combat, of course). It's pretty bare bones but the variety is appreciated. There are lite-RPG dialogue/story mechanics and quests, though I do not want to give the impression this narrative heavy game. It's an RPG style that used to be relatively common but not so much anymore.
But the real highlight is the New Horizons mod which greatly overhauls the game. It's been developed for almost 20 years. I don't recommend playing the vanilla game - I enjoyed it at the time, but it's just an inferior experience to the mod.
Best yet, it's free. The game is abandonware.