You are correct that the configuration here is not a parallel plate capacitor. It is a more general sort of capacitors. One in which the two plates have been "unfolded" so that their areas are actually in the same plane.
Imagine a charge parallel plate capacitor with electric fields passing from one plate to the other. Now imagine tilting the two plates away from eachother. There will still be an electric field passing from one to the other, it will just have curvature. Now imagine continuing to tilt the plates away from eachother until they lie in the same plane. There will still be electric field lines running from one plate to the other, it will now just be "looping through" the area in between the two plates.
That is, the charge on the two plates creates an electric field which loops through the space in between the plates. One would have to do work to move a charge against those electric field lines from one plate to the other.* Therefore, for a given charge $Q$ on the plates there is a corresponding voltage $V$ between them.
This is a fundamental property of a capacitor. The basic idea is, that for a given conductor geometry, if you charge up the conductors with equal and opposite charge you will set up a particular electric field pattern. This electric field pattern is associated with a voltage between the two conductors. We call the ratio of charge to voltage the capacitance of the conductor geometry.
Ok great, with that out of the way we can now see how a stud finder works. Essentially the key is that if a dielectric is placed anywhere within the region where the two conductors create an electric field that dielectric will serve to decrease the electric field in the region of space thus decreasing the voltage which arises for a given charge and increasing the capacitance. Imagine calculating the voltage by adding up the $\vec{E}$ field moving along a path that goes through the dielectric.
By monitoring the capacitance in real time the electronics in the stud finder are thus able to detect the presence of a large volume of dielectric material close to the device which is interpreted as the presence of a stud.
A few tricky points, remember (or learn), that (in electrostatics) the voltage between two points in space can be found drawing any path between the two points and adding up the contributions of the electric field parallel to that path at all points. This tells you why you will get the same answer whether you take a looped path through the dielectric or just a straightline path through air between the two conductors.
*Alternatively we could say that the integral of $V = \int\vec{E}\cdot d\vec{l}$ along a path from one plate to the other is non-zero.
