Shifting a lens vertically, laterally or to some intermediate angle changes the angle of view framed by the recording medium. Lenses with shift function project a relatively large diameter image circle - shifting the position of the lens or the recording medium changes the portion of the image circle framed by the edges of the media. Pointing the camera in a different direction has the same effect. This is simple geometry and has nothing all to do with stitching multiple frames together. Tilting the lens or the film/sensor plane can change the framing of the subject as well.
That's correct, unless one offsets the displacement of the entrance pupil by a counter movement of the whole camera+lens. That's what the setup that I posted a picture of earlier in this thread achieves. It compensates for the horizontal lens shift, a common setup e.g. for landscape photography with some downward tilt included.
Indeed, tilting the lens will also introduce an offset, but it's only relevant if the amount of tilt is changed
between shots, and the amount of off-axis displacement can be relatively limited, and can sometimes be hidden in the overlap blend. On my 1Ds3, the TS/E 45mm e.g. shifts the image center projection roughly 37.5 pixels (0.24 mm) per degree, the TS/E 90mm some 13.8 pixels (0.09 mm) per degree, and the TS/E 24mm II some 36.1 pixels (0.23 mm) per degree of tilt.
A good panorama stitcher can include these offsets (if not already compensated for when shooting) for an improved registration of the image tiles in a rotational stitch, but if only pre-compensated lens shifts are involved then stitching becomes basically simple overlaying of shifted images, even Photoshop can manage that.