Where is the mistake in this solution of $\lim_{x \to 1}{\frac{1-x^2}{\sin (\pi x)}}$?

Your third equality attempts to make use of the rule $\lim\limits_{x\to0}\frac{x}{\sin x} = 1$, but note that yours has $y\to 0$ yet the argument is not $y$, it is $\pi(y+1)$, which does not go to zero. That's where your work goes wrong.

$$\lim _{ y\to 0 }{ \frac { \pi (y+1) }{ \sin (\pi (y+1)) } } =\frac { \pi }{ 0 } \neq 1\\ $$