Why is $|x| \neq x$? I was just thinking about the absolute value function. Why does the following equality not hold?
$$|x| = \sqrt{x^2} = (x^{2})^{1/2} = x^1 = x$$
After all, there are clearly some values of $|x|$ that are not equal to $x$. Is this a domain/range issue?

Because your third equality is not true. If you take the square root of a square, you must include absolute values because the square has already mapped the function to the positive reals, which is why you do not have a domain issue with the square root in the first place.
