Let's say I have an picture of the sky or of an landscape and I want to measure an angle in the celestial sphere. Is there any software for that? In real life I would use the empirical hand gestures or a theodolite.
A plate solver matches the stars in an image to a catalogue of star positions. Astrometry.net provides a web service and a software API for this. Given only another user's cell phone photo, Astrometry.net determined that the image scale was 112 arcseconds per pixel. It also estimated an image size of 72 by 54 degrees, apparently using an approximation which is only valid for small angles.
Assuming low distortion (not a fisheye lens), the image is a gnomonic projection of part of the celestial sphere. If $r$ is the distance from the image center in pixels, $p$ is the pixel scale in radians per pixel, and $\theta$ is the angular separation from the center in radians, then
$$rp = \tan \theta$$
For example, I used the measurement tools in GIMP and Stellarium on two bright stars near the center to verify the central image scale of 112 arcsec/pixel or 0.54 mrad/pixel. The edges of the 2320x1740 image are 1160 and 870 pixels away from the center. Solving for $\theta$, the edges are 0.56 and 0.44 radian away from the center, so the image's angular dimensions are 64 by 50 degrees from edge to edge.
Daytime images would of course have the same pixel scale and angular dimensions as astrophotos taken with the same equipment.