Sail tents are called sail tents for the simple reason that they look like a sail. That said a ship does have a cloth called the covers, and which in heavy rail and whilst docked is tied from gunwale to gunwale over the top of the lowered yard with the reefed sail under it. This keeps rain water out of the ship which would only need bailing out later. It also protects the sail. This practice is carried on to this day on modern yachts, though it is the ship's covers that is used for the onshore impromptu shelter that we call the sail tent.
A huge misconception is that the sail tent is actually the sail from a ship. This is almost certainly completely wrong. The sail is the ship's engine and in wealth terms they cost an absolute fortune. They were so prized that the only finds we have of early sails were found in a church, presumably put there for safe keeping. There is no way any captain is going to allow his precious sail to be used as a common shelter over a fire. Furthermore to get the sail on and off the yard can take several hours, so in practical terms you are not going to do this.
It is highly unlikely that people would cut local trees to form poles for tents etc. This is perfectly feasible for upright support poles, but completely impractical for crossbars and other structure that comes into contact with the cloth. This is because the cloth moves with the wind and will quickly snag and tear on any irregularity that it is in contact with. From experience even a loose cloth that constantly flaps against something will be damaged pretty quickly. It follows that anyone intending to put up a tent would at least have a smooth prepared crossbar or two, and in all probability there being no guarantee that there will be suitable other poles available at your destination you would carry vertical support poles with you as well.
Credits: with thanks to Steve Lines.