The advertised so called field of view (FOV) for a telescope eyepiece is the “apparent field of view”. To get the “actual field of view” it is necessary to divide the apparent field of view of the eyepiece, the secondary, by the magnification. This will vary according to the primary focal length of the telescope.
With respect to full frame and crop frame DSLR cameras, the combination of primary and secondary lenses are combined in the camera lens. A principal design constraint is to produce an image circle adequate to cover a 43.3mm diagonal. Consequently, the magnification of any particular lens will be the same. Any given 50mm lens is typically regarded as 1x, and a 400mm lens as 8x. The actual field of view will vary simply as a factor of the focal length.
When a 400mm lens, of 8x magnification, is used on a full frame or crop frame sensor the FOV of the lens is identical. The image circle produced by the lens is designed to adequately cover the diagonal of the full frame sensor. In the case of the crop frame sensor, the image circle simply splashes over the edges and is wasted. Like an eyepiece capable of an 8mm exit pupil for an iris of your eye which might only open 5mm.
The crop sensor suffers a disadvantage of being a small sensor inside a large image circle. So a 300mm lens on a crop sensor body will provide a narrow FOV similar to that of recorded by a 450mm lens on a full frame body. The crop sensor camera only sees the centre portion of the image circle. At no time do you get the magnification equivalent of a 450mm lens on your crop sensor body. The magnification factor is identical for full frame and crop sensor.
Lenses designed for crop sensor cameras, such as Nikon's DX line or Canon's APS-C line, should produce a smaller image circle to fit a smaller sensor format. This would reduce the image waste, and potentially concentrate the available light on the sensor by about ½ stop. However in my experience I have not seen the extra half stop of exposure advantage, so I suspect that these lenses do not properly produce a significantly smaller image circle.
Also bear in mind that “full frame” varies with the native format of the camera. 4X5” view cameras are full frame with their particular lenses, 6×6 Hasselblads are full frame with their lenses, and 4/3 cameras are full frame with their lenses. Each format has a lens design suited for the diagonal of their format. And each has similar FOV with a different focal length lens.