Shot or thermal noise of a MOS transistor
You cannot always assume that a current \$I\$ flowing through a device will result in shot noise. You need to understand that the shot noise arises when a current flows through a potential barrier like from p-doped region to n-doped region in a pn-junction diode.
For a MOSFET in strong inversion region, there is very small potential barrier between source/drain and the channel (because of the positive gate potential) consequently the noise is mostly thermal. In a weak inversion region the potential barrier is higher and in this case, the noise would behave as shot noise.
Also, note that shot noise and thermal noise are related to each other, in the sense that they have similar origins. So if a current is passed through a region with potential barrier, then the noise behaves as shot and if the potential barrier is reduced to zero, the same noise behaves as thermal noise.
Refer to:Unified derivation of Johnson and shot noise expressions, AJP (2006).