Step-by-Step Solution:

Option A:

- $f(x) = x$ has a domain of $\mathbb{R}$.
- $g(x) = \frac{x^2}{x}$ simplifies to $g(x) = x$ for $x \neq 0$.
- Therefore, the domain of $g(x)$ is $\{x | x \neq 0\}$.

Since the domains of $f(x)$ and $g(x)$ are different, they do not represent the same function.

Option B:

- For $f(x) = \sqrt{x^2 - 1}$, the domain is determined by $x^2 - 1 \geqslant 0$, which simplifies to $x \leqslant -1$ or $x \geqslant 1$.
- For $g(x) = \sqrt{x - 1}\sqrt{x + 1}$, the domain is determined by $\left\{\begin{array}{l}{x - 1 \geqslant 0}\\{x + 1 \geqslant 0}\end{array}\right.$, which simplifies to $x \geqslant 1$.

Since the domains of $f(x)$ and $g(x)$ do not completely match, they do not represent the same function.

Option C:

- $f(x) = x$.
- $g(x) = \sqrt{x^2}$ simplifies to $g(x) = |x|$.

Since $f(x) \neq g(x)$ for $x < 0$, they do not represent the same function.

Option D:

- $f(x) = |x| = \left\{\begin{array}{l}{x, x \geqslant 0}\\{-x, x < 0}\end{array}\right.$.
- $g(x)$ is defined as $\left\{\begin{array}{l}{x, x \geqslant 0}\\{-x, x < 0}\end{array}\right.$.

Since both $f(x)$ and $g(x)$ have the same domain and correspondence rule, they represent the same function.

Conclusion:

The pairs of functions that represent the same function are given in option D.

Therefore, the correct answer is $\boxed{D}$.