>Why can't I use $I_1\omega_1=I_2\omega_2$ in this case? Since $I$ was taken of the entire system in the first problem. So why can't I just take $I$ of the system in this case? Which would obviously give the equation $0=\left(\frac{MR^2}{2}+mR^2\right)\omega$ where $m$ is the mass of the person. 

So what are you going to plug in for $\omega$? The man and the disc have different angular velocities. If we consider the man is moving clockwise, the disc is rotating counter-clockwise. Therefore they don't have a common $\omega$.

But in your first question, all bodies have the same angular velocity.

Actually your second equation is not incorrect. It will give you $\omega=0$ which is the angular momentum of the *system*, as expected. Though you are asked to find the angular velocity of the disc. Therefore you have to consider the components of the system (man and disc) seperatedly.