That video gives you an idea of Saddleseat equitation. Saddleseat however does cover many classes. The riding style is the same but, there are 5 gaited and 4 gaited. Pleasure and Park horses. So yes, it is a style but there is variation in what you might wish to do. This generally depends on your horse and where his/her talent lies. There are also many different breeds that are ridden saddleseat.
This one is of the 2012 five gaited world championship. Not the best video as it seems to have been taken from the nose bleed section but, gives you an idea of the fun in the class. High energy and a blast to ride. Note that much of the equitation falls by the wayside here. This is about showing the horse to it's very best.
Here are 3 gaited park horses. Very animated, classy.
Some other Saddleseat breeds.
There are many other breeds that go Saddleseat, including the Kentucky Mountain horse. This just gives you an idea of what it looks like. Fun to ride, high energy and exciting. Good luck to you.