Oh boy, we do no such thing. You just haven’t been taught it. Division by zero is defined in a set of mathematics called Wheel Theory. Look it up, you sound interested.

The problem is that this generalization causes a loss of generalization for other properties. Such as the distributive law, the existence of unique multiplicative inverses etc. While it is useful to look at algebraic wheels and see what is true about them, we can prove a lot more if we stick to fields. That’s why people prefer to use them.

It’s wrong, flat out wrong to say “you can’t divide by zero” but mathematicians say it all the time. Why? Because the algebraic system you develop is much more useful. Dividing by zero is of much smaller utility than the distributive law, for example. But it would be more correct to say “you can’t divide by zero, if you want the distributive law to hold”

Edited to add that 1 =/= 2 but only because 0=/= 1 by definition. This is again due to a frustrating loss of generality would result. There are however number systems where 0 = 2 (modulo arithmatic). Or even crazier things like apperently infinite numbers being equal to negative numbers (p-adic numbers).