Before I give away the punchline, try to imagine the orbit in a reference frame centered at the location of the black hole. Obviously, the motion should be elliptic; lets define a Cartesian coordinate system such that, say, the x-y plane coincides with the plane of the elliptic orbit. Now, we know that the black hole should reside at the focus of this orbit. But, as a far away observer, what are the chances that you would be seeing the orbit exactly face-on - i.e., you are sitting on the z axis, just really far away?

So, yes, the black hole is at the focus of the elliptic orbit, not the focus of the ellipse formed from the projection of the orbit onto your plane-of-sky, which geometry tells us is also an ellipse.

BTW, this isn't a problem for the astronomers. You can back out exactly what that inclination of the orbit is, provided you can measure the line-of-sight velocities of the star. (Actually, relative velocities will do; no absolute calibration necessary.) This is easily done by looking for some distinctive stellar spectral features and noting their relative shifts at different parts of the orbit. From this you can reconstruct the orbit completely. The key scientific results they derived was the even subtler redshift of these spectral features as the gravitational field grew stronger when the star approached the black hole.

edit: actually, you could do even better, and use the offset of the black hole from the elliptic focus, and use projective geometry to get the inclination angle!