I know that unsalted butter has a water content - can I substitute clarified butter, vegetable oil, or shortening in equal amounts? If I do use unsalted butter, do I need to adjust the hydration percentage? Adversely, if I use a fat with no water content, do I need to adjust the hydration level up a bit? Or do I just rely on how the dough comes together, and tweak as I go?
If you're the type of person who measures all your ingredients on a gram scale, theoretically you do.
However, unless the fat in the recipe is on the order of 1/2 cup, the amount of omitted water is going to be less than the amount of variance in water you get from uncontrollable sources, such as the ambient humidity. There's also that you never know exactly how much water is in that generic unsalted butter to begin with. American butter varies between 11% and 18% for water content, not just between brands, but also by season and year as the price and quality of cream fluctuates. Not only do you not actually know how much water content is in your butter, you don't know how much was in the cookbook author's butter (also, some cheaper brands of ghee, such as the one sold at Trader Joe's, are not actually water-free).
If we were to assume that a recipe called for 200g of unsalted American butter, then you'd theoretically want assume an average water content of 15%, and swap that for, say, 170g of oil and 30g (roughly 2 Tbs) of water.
But my practical advice is "add a couple extra teaspoons of water and see how it comes together".
In their analysis of Chocolate Chip Cookie recipes America's Test Kitchens alternated butter, shortening, egg yolk and some other oils. They described various subtle differences between the results (flatter/fluffier/etc) based solely on the fat substitution. All of the variations were described as "good" but (in their test case) they settled on adding one additional egg yolk to their recipe (This is the cookie recipe I use...great, thanks, now I'm going to have to make cookies tonight).
Ultimately your answer comes down to what is important to you. Different fats (and as FuzzyChef points out - hydration levels) are going to result in different tastes and textures, but they will all likely end up in the range of "good" but may not be "the best". (none of them should explode) Will it be 'the same'? No.