Use Menu Processing / Toolbox / Minimum bounding geometry / Minimum Oriented Rectangle. Than the values you look for correspond to half of the sited of the rectangle.
Edit: as described in @Kübras solution, the
Minimum Oriented Rectangle already contains an attribute for width and height - thus the following step, desribed in the original answer, is not necessary.
To calculate the sites, convert the polygon to lines and explode the lines - this last tool will automatically create an attribute for width and height.
Screenshot: I created the ellipse with this expression:
make_ellipse( make_point (0,0), 22, 55, 72) - thus 22 and 55 for semi-minor/semi-major axis, 72 for azimuth. This corresponds almost to half of the values for the width/height attributes - the ellipse is not really round an contains just the default number of 36 segments, hence the rounding error: