Why does the closest approach of star S2 to Sgr A* not appear to be near the focus of its elliptical orbit?
The orbital elements are on wikipedia:

$$e=0.884\ a=0.125'',\ i=134^\circ,\, \Omega=228^\circ$$

(At an assumed distance of 8kpc, $0.125'' = 1000au$)

It is the inclination that means that the black hole is not at the focus of the projected ellipse.

Imagine a circular orbit, with a central body.  If viewed from a distant but high inclination, the orbit is projected as an ellipse, with the central body at the centre of the ellipse (not the focus) 

A distant projection of an ellipse is another ellipse, but the projection of the focus is not the focus of the projection.

Below you can see another example.  The ellipse is in the grey plane, with foci at A and B.  But the apparent ellipse has a completely different axis, and the focus is nowhere near the axis.  These image have been generate with geogebra and no perspective has been used.