Confused about this KVL equation (AC circuit)
If I understand correctly, they used loop analysis: with one loop around the entire circuit and one loop around the right mesh (Io).


I'm not sure I see the second loop in this analysis. It looks like the first three equations are just Ohm's law - the loop around the right mesh never really comes up (if it did, you would see a KVL equation for it).

In any case, the current \$I_0\$ is literally defined as the current through the capacitor, so the voltage across it is
$$
V_{cap} = IZ = I_0 (-j1 \Omega)
$$
If you replaced it with two loop currents (as you described), then the current through the capacitor and the 1 ohm resistor would be \$I_1 + I_2\$, and you'd have to re-consider the first three equations.