As with a lot of other things, the trick rests in the nuances of the definitions.

Force is a well-defined concept. What it means is an interaction between two or more objects which contributes to a change in their momentum. The quantity of force is defined as the vectored size of this change:

$$\mathbf{F} = \frac{d\mathbf{p}_\mathrm{contrib}}{dt}$$

. Now this statement also carries over to quantum mechanics. The difference is momentum is now an operator on the Hilbert space, $\hat{\mathbf{p}}$, instead of a simple vector. But we can still define the quantity of force, $\hat{\mathbf{F}}$, in the same way, via the Heisenberg picture in which it is the operators that are changing and not the quantum state vector. Note that the momentum operator doesn't always change! For an isolated system where momentum is conserved, its non-doing so is exactly how conservation of momentum expresses itself in quantum theory.

The Pauli exclusion principle is a principle regarding how a joint state vector of a composite system of a certain kind of identical particles is to be built up from those of the individuals. As it only is talking about state vectors, it does not imply any change in the operators, thus by implication no change in the momentum operator, and thus is not a force.

The normal force does indeed have something to do with the PEP, but it's more like a "team job" between both the PEP and electromagnetism. PEP sets a limit, and electromagnetism provides the actual force - the $\frac{d\hat{\mathbf{p}}_\mathrm{contrib}}{dt}$ - to "enforce" (:D) it.

As with a lot of other things, the trick rests in the nuances of the definitions.

Force is a well-defined concept. What it means is an interaction between two or more objects which contributes to a change in their momentum. The quantity of force is defined as the vectored size of this change:

$$\mathbf{F} = \frac{d\mathbf{p}}{dt}$$

. Now this statement also carries over to quantum mechanics. The difference is momentum is now an operator on the Hilbert space, $\hat{\mathbf{p}}$, instead of a simple vector. But we can still define the quantity of force, $\hat{\mathbf{F}}$, in the same way, via the Heisenberg picture in which it is the operators that are changing and not the quantum state vector. Note that the momentum operator doesn't always change! For an isolated system where momentum is conserved, its non-doing so is exactly how conservation of momentum expresses itself in quantum theory.

The Pauli exclusion principle is a principle regarding how a joint state vector of a composite system of a certain kind of identical particles is to be built up from those of the individuals. As it only is talking about state vectors, it does not imply any change in the operators, thus by implication no change in the momentum operator, and thus is not a force.

The normal force does indeed have something to do with the PEP, but it's more like a "team job" between both the PEP and electromagnetism. PEP sets a limit, and electromagnetism provides the actual force - the $\frac{d\hat{\mathbf{p}}}{dt}$ - to "enforce" (:D) it.