So in this fantasy universe the dominant maritime powers developed & embraced multi-hulled ships akin to Polynesian catamarans very early in their history. They've made the same developments in weaponry, navigation, ship building, etc. that Europe had by 1800, but with multi-hulled designs rather than single-hull.
Do multi-hulled ships cause any significant change to how naval battles are fought or how naval warfare is conducted?
I'm not a naval architect but I see some serious issues.
Firstly Economics: conventional merchant ships (18th century or not) are capable of carrying far more tonnage and being far larger than catamarans ever could. That same principal also applies to warships, they can also carry more tonnage - in this case cannons and manpower. So a catamaran faced with fighting a conventional ship of the line would always be out gunned and out manned. Its only real advantage being speed. It also means that in boarding actions the crew of a cat would have to fight their way 'up' onto a ship of the line while their opponents would be fighting their way 'down' which would be far easier.
The second issue is building materials. You are restricted to timber. This stickily limits the size you can build cats without structural integrity issues arising. This also feeds back into point one above. Conventional hulls could withstand much more battle damage than cats. Same applies to storm damage, all ships flex to dissipate the strain of riding over/through a wave. Beyond a curtain size this gets harder for cats because you have two hulls linked by timber spars which are also under strain. And then there's the cost. Even if you could build cat approaching the size of a normal warship it would be more expensive to build because it would be far more complex and need much bigger yards.
Also note that Polynesian cats did not use the space between the main hull and the outrigger as a fighting/cargo platform. That's because this space had to be left open to the sea or waves would break over it - which also puts more strain on the support spars. Remember if those break the cat is doomed, the hull is not stable without the outrigger. They were also in part oar powered and could be because they rode so low in the water.
Where they might, I suppose be able to perform a useful role in an 18th century setting would be as scouts or courier vessels due to their speed - if you could work out a suitable sailing rig. Perhaps a lanteen sail 'dhow' style rigg (on 1-3 masts) might suffice but I'm not a sailor so that last part is pure speculation on my part
They can have advantages
First things first. They can carry less weight and are less efficient in storing much stuff. You also need more space in the water for the boat to work. This is a detriment, as you cannot get as many men and cannons on a multi hulled boat than a normal big one. Finally, multi hulled ships generally are unable to turn as well as a single hulled boat, again because of drag in the water.
That being said, they can have advantages as well. Multi hulled ships in modern times are made for their lower water resistance. Especially at speed they can rise higher in the water, decreasing drag and thus increasing speed. Multi hulled ships can also be created to have a lower profile.
Speed and a lower profile can be a huge advantage. These two properties make them more difficult to hit, while able to fire back easily, with more easy hits close to the waterline of the enemy ship. The enemy is also at disadvantage, as due to your speed you can determine most of the engagements. You can decide to stay out of range or in it, making sure the enemy can neither escape or easily capture you.
To further take advantage of this, you can make multi hulled ships smaller and thus better in manoeuvrability. This will maximise the amount of damage you can do while at the sides that have few or no cannons, like the front and the back of the enemy ship. In addition, you can get multiple people onto the enemy ship by gaining access via the back.
What you lack in direct firepower you'll have in staying power and ability to determine fights. Do note that this isn't a perfect strategy and is best done with multiple ships. Against an armada they will have their advantages diminished. This doesn't mean that they cannot win. If memory serves, a Spanish armada that was called unsinkable was nearly destroyed when they went to the Netherlands to right some slight. The Dutch used many lower fast ships to manoeuvre out of the worst area's of the enemy ships and then fire at or board them.
Raw firepower isn't everything. Multi hulled ships can be used as both scouts and quick but possibly effective ships against much bigger enemies.
It pretty well won't happen because of physics.
Multi-hulled boats let you get the width of a large ship for stability with the draft of a small ship for speed and shallow-water handling. They're also easier to construct per unit width.
But just the barrel of a 4 pounder cannon -- the smallest one might seriously expect to use for naval warfare -- weighs 600lbs. A more typical 8 pounder had a barrel closer to 1200 pounds. And these were small guns, most commonly used by merchant ships for self-defense. Real warships were armed with up to 42 pounders. That's a 7 inch bore diameter! And 42 pounds per cannon ball, let alone the actual gun!
Warfare in the 18th century needed the carrying capacity of a deep-draft, massive ship. And by the time you get that much capacity, physics makes it much, much cheaper to build it as a single hull.
When you double up your hulls in a multi-hulled design, you get double the carrying capacity. Great...
When you double the volume of a single hull though, it takes way less than double the materials to build.
This is why we use massive super-freighters nowdays instead of fleets of smaller ships. It's just more efficient in a lot of ways.
So the only way you're going to see naval warfare with cannons using shallow-draft, multi-hulled designs is if there's some external factor forcing it. Maybe if the sea isn't that deep, or has a lot of shallow areas at least so they can't have a deep draft. Of course that'll change all manner of tactics about being able to do things with the anchors and easily recover sunken cargo. It'll even change the characteristics of storms.
The smaller carrying capacity will mean smaller, shorter-range guns. So tactics, maneuvering, and boarding capabilities will be more important. Additionally, the higher cost of weight will make it more economical to use fewer, more expensive guns instead of larger numbers of cheap ones, so things like the Puckle Gun are more likely to actually catch on and be widely used.
Multi-hulled ships were used for hosting giant catapults in the Hellenistic age: http://www.hellenicaworld.com/Greece/Technology/en/GiantShips.html
However, the line tactics developed in order to maximize firepower is by exposing as many guns on the side of the ship as possible. That was the reason behind having multiple rows of guns, and large displacement to compensate for the weight of the guns and ammunition. The multihulled ship would have a disadvantage in firepower, since side-to-displacement ratio is smaller.
The best way for a multihulled ship to maximize firepower would be to allow all guns to fire in all directions. This might be difficult in a sail-driven ship, which would have masts, stays, spreaders, booms and ropes.
If we look at the history of ships from 800 to 1700, we see the ships getting taller and taller. The Viking long ship was quite low and long. Why get taller? So that they could fire down into the enemy ship. And so that it was hard for the pirates in smaller boats to climb over the sides. Notice how the Portuguese carrack (1400-1550) is almost a "U" to have platforms for fighting down at the enemy.
The multihulled ship is a design for fast, lightweight attack and get out of there. The Viking long ship has the same advantage. Look at why it was given up. Put a platform on another ship and the long ship is at a disadvantage.
You are suggesting that they give up those important advantages but are not offering a good alternative.
