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I believe putting it this way could be helpful:
One one hand, $\frac{x - 2 }{3 x - 6 }$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 viewed as a function that needs to accept ANY input and provide you with $\frac{1}{3}$back. In particular, the domain restriction at $x = 2$ prevents $\frac{x - 2 }{3 x - 6 }$to be the same thing as $\frac{1}{3}$
Hence, simplifying wouldn't be appropriate $\frac{x - 2 }{3 x - 6 }$to $\frac{1}{3}$unless we are working in a domain where $x \ne 2$

Yes... and no.
It is impractical to properly lay out all of the subtleties of what one intends with math notation, which is ambiguous. We've had ages to learn how to create notations where ambiguities frequently don't matter but do in the fine print.
There are several distinct interpretations of what $\frac{x - 2 }{3 x - 6 }$; The status of "evaluation" is where the alternatives diverge most noticeably at " $x = 2$".
The most basic interpretation is that $\frac{x - 2 }{3 x - 6 }$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.
There are also more ways to interpret "take the continuous extension": essentially, you take the function's graph and fill in all the gaps; here you would add in $( 2 , 1 / 3 )$. Also, if you were using the extended real numbers you would add in $( + \mathrm{\infty } , 1 / 3 )$ and $( - \mathrm{\infty } , 1 / 3 )$, so that evaluation at $\pm \mathrm{\infty }$ 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
$\frac{x - 2 }{3 x - 6 }= \frac{1}{3}\cdot \frac{x - 2 }{x - 2 }= \frac{1}{3}\cdot 1$
The witnesses do "cancel" to depart if we adopt one of these "continuous extension" interpretations $1 / 3$