Isn't that a circular definition, though? sqrt(x²) = ±x, which isn't a function since there are two values in the domain for each value in the range (other than zero). The version they're equating with abs(x) is the absolute value of the square root, or abs(x) = abs(sqrt(x²)), which is true but wouldn't help prove that abs(x) is a function.
Of course, the underlying problem was the premise that a function must be defined by exactly one formula.
Why not just say that abs(x) = ±x, ignoring the negative solutions? If you'll accept "the non-negative square root of x²" then I see no reason to reject "the non-negative component of ±x". Both are single formulas with positive and negative solutions combined with a qualifier rejecting the negative solutions.
When people write "sqrt", they mean the function, or, "principal square root". "A square root" is a different thing. Saying "the definition" makes sense only within the context where the definition is established, otherwise we have to fallback to the common usage. Wittgenstein would laugh at this conversation.
Also, I'm not arguing for this convention ("it needs to have a single formula to be a function"). And when you start substituting symbols, almost nothing withstands this test. Because then you can always always twist the formula into several cases. And generally thinking about functions as something that is defined by formulas is a very limiting view. Almost all functions cannot be defined by formulas.
At the point when there was an effort to formalize these things ideas like RPN starting popping up.
Mathematics also deals with a much more flexible medium. It's written on a plane, in 2 dimensions, whereas programmers confine themselves to just one. Mathematicians are also free to make up their own notation and symbols, whereas programmers are confined to a single alphabet and most often can't extend the syntax.
EDIT: Case in point: some mathematicians argued for a while if abs (absolute value) is a function. Some argued that it's not, because you need two formulas to define it. When somebody pointed out that (for real numbers) it's just "sqrt(x^2)", the first ones agreed. Sounds silly in retrospect now that we have a set-theoretic definition, but at the time it wasn't obvious and all these functions looked very different from each other.