Vertical?
There was a discussion about 2 years ago where the consensus was that FOV is the vertical extent.
Advanced camera: Horizontal
I think that Advanced Camera Tools are treated differently, however.
I made a sample scene with a spatial reticle doohickey in it and placed a camera at the apex. When looking through that camera, there is an unambiguous angular scale in view which also serves as position check for the camera.
I chose the camera type to be full-frame still camera, 36 mm width, aspect ratio 3/2 = 1.5. In the camera’s scene, I modified the focus (Opt + Up or Down on Mac) to get a FOV of 45° vertically, using one of the rings in my reticle doohickey. At that point, ƒ is reported as 29 mm. Basic trigonometry confirms that this is consistent.
Choosing Camera → Field of View at this time will populate the Measure field as “63.65 deg.”, which happens to agree with the FOV as read from my reticle along the horizontal axis:
I repeated the exercise by choosing the vertical FOV to be roughly 30° by setting ƒ = 45 mm by Opt+Up/Down in the camera view, after which Camera → Field of View reports “43.60 deg.” Therefore, this appears to be the intended behavior.
The horizontal sweep angles of the frustum volume also agree with the reported FOV.
There appears to be a mild inconsistency: The reported FOV does not change when changing the Aspect Ratio in Edit Camera. 
It appears a fixed aspect ratio is used to report FOV, perhaps the ratio implied by the camera type chosen from Select Camera Type.
Regular camera: Vertical
Alas, now choosing Tools → Advanced Camera Tools → Reset Camera will zoom out and create a view with indeed 63.65° vertical FOV, confirmed by the reticle:
Dolly Zoom
There is another indication that the regular SketchUp camera is treated differently from Advanced Camera Tools, namely, the Camera → Zoom tool performs a dolly zoom. It breaks out of an Advanced Camera view (if active), and clearly does not alter the focus length of an Advanced Camera as one might expect. SketchUp’s Zoom action being a dolly zoom always alters the camera position.
I have a feeling there are historical reasons for the behavior of the two camera concepts.