Why $s/(1-s) = 1$ at $s=1$ in bode plot? Wolfram plot of $\frac{s}{1-s}$ is $\pm\infty$ at $s=1$. But, bode plot of $\frac{s}{1-s}$ results in $1$ at $s=1$. Obviously, this is wrong. Why?

The x-axis in the Bode plot isn't $s$, but $\omega$. Remember that $s=j\omega$, so what you're seeing in the Bode plot is as follows:

$$20\log|H(j1)|=20\log|\frac{j1}{1-j1}|=20\log|\frac{j(1+j)}{2}|$$