Does $\frac{x-2}{3x-6}$ really equal $\frac{1}{3}$?

I think it may help to frame it as this:

One one hand, $\frac{x-2}{3x-6}$ is a function that is defined on $\mathbb{R}$ except at $x=2$, and at every point it is defined it is equal to $\frac{1}{3}$.

On the other hand, $\frac{1}{3}$ can be seen as a function that must take in ANY input and give you $\frac{1}{3}$ back. In particular, the domain restriction at $x=2$ prevents $\frac{x-2}{3x-6}$ to be the same thing as $\frac{1}{3}$.

Thus, It would not be correct to simplify $\frac{x-2}{3x-6}$ to $\frac{1}{3}$ unless we are working in a domain where $x \neq 2$.

Yes... and no.

Math notation is ambiguous; it is impractical to fully spell out all of the fine details of what one means. We've had centuries of wisdom in developing notations where the ambiguities usually don't matter, but they often do in the fine detail.

There are a few different things one might mean by $\frac{x-2}{3x-6}$; the most significant disagreement between the alternatives is the status of "evaluation at $x=2$".

The most basic interpretation is that $\frac{x-2}{3x-6}$ is a recipe for performing a sequence of arithmetic operations upon a given input; in this interpretation, evaluation at $x=2$ is, in fact, undefined.

Many other interpretations can be described as "take the continuous extension": roughly speaking, you take the graph of the function and fill in all of the holes; here you would add in $(2,1/3)$. Also, if you were using the extended real numbers you would add in $(+\infty, 1/3)$ and $(-\infty, 1/3)$, so that evaluation at $\pm \infty$ is defined. (similarly if you were using the projective real numbers)

Your teachers description is nonsense when taken literally; however, the likely intention is that he is using "$0$" as a stand-in for some sort of 'witness' of vanishing; e.g. we factor out $(x-2)$ from both the numerator and denominator to get

$$ \frac{x-2}{3x-6} = \frac{1}{3} \cdot \frac{x-2}{x-2} = \frac{1}{3} \cdot 1$$

If we use one of these "continuous extension" interpretations, the witnesses do 'cancel' to leave behind $1/3$.