I'm confused about whether the same focus length across different lenses on the same camera always produces the same Field of View. Some lenses are advertised as wide-angle and some are not. Is there a difference in the FOV?
A certain focal-length given a fixed sensor-size is expected to show the same field of view. That is, on the same camera, two lenses of the same focal-length will give the same field-of-view. There are two catches however:
If two lenses have exactly the same focal length and exactly the same amount of distortion, then on a camera they have the exact same field of view.
The exception is when you get to very wide angle lenses and some may be fisheye lenses while others are rectilinear. In that case, they will have different fields of view.
You will find that you cannot buy two lenses with the exact same focal length and exact same amount of distortion, so there will be small variances in FoV between lenses of the same focal length.
The camera lens projects an image of the outside world onto the surface of film or digital image sensor. Focal length is a measure taken when the lens is imaging a far distant subject like a star; this is a distance we call infinity (as far as the eye can see symbol∞. This measure is from a point called the rear nodal to the image plane (focal plane).
The size of the projected image and the angle of view is a function of the focal length of the lens and the dimensions of the film frame / digital chip size (format). For example, the image size projected by a 100mm lens will be twice as large as the image projected by a 50mm.
It is customary to fit a lens to a camera that is approximately equal to the diagonal measure of the format. Such a lash-up produces a diagonal angle of view of 53. As an example, the 35mm full frame measures 24mm height by 36mm length. The diagonal measure is 44mm. Mount a 44mm and the angular field will be 30⁰ vertical - 45⁰ horizontal - 53⁰ diagonal. It is the diagonal value that is most often given. I find this value nearly useless. This falls in the same category as TV sets which are advertised by the diagonal measure.
Table of angle of view some selected focal lengths mounted on a 35mm full frame Fx.
25mm 51⁰ Vertical 72⁰ Horizontal 82⁰ Horizontal
28mm 46⁰ Vertical 65⁰ Horizontal 75⁰ Horizontal
35mm 38⁰ Vertical 55⁰ Horizontal 63⁰ Horizontal
44mm 29⁰ Vertical 43⁰ Horizontal 53⁰ Horizontal
50mm 27⁰ Vertical 39.6⁰ Horizontal 47⁰ Horizontal
85mm 16⁰ Vertical 24⁰ Horizontal 29⁰ Horizontal
105mm 13⁰ Vertical 19⁰ Horizontal 23⁰ Horizontal
135mm 10⁰ Vertical 15⁰ Horizontal 18⁰ Horizontal
200mm 7⁰ Vertical 10⁰ Horizontal 12⁰ Horizontal
400mm 3.3⁰ Vertical 5⁰ Horizontal 6⁰ Horizontal
Angle of view = 2arctan d/2f
d=dimension of the format = width of Fx frame = 36mm
f = focal length = 50mm
Compute angle of view for 50mm mounted on Fx camera 36mm length
Angle of view = 2acrctan d/2f
Angle = 2artan 36/100
Angle = 2artan 0.36
Angle = 39.6⁰
You've already gotten good answers on minor variation between lenses of the same stated focal length due to rounding and distortion, which answers the broader issue, but specifically as to "some are labeled as wide angle and some art not" the answer is "no, there are not major differences, this is simply about marketing choices."