Is there a common name for the surface z = xy? I would call it a saddle, but it's not the standard saddle. Is there a standard name for it, the way we have 'hyperboloid of one sheet' for example?

Saddle point is an attribute/ character of all surface points. If you hold a 3D model of it in your hands, its nomenclature is invariant by the direction of your view :)  but lines of projection can have a separate name.
EDIT 1:
Sorry did not follow OP properly at that time, (now deleted phrase hyperboloid of 1 sheet). All points of negative Gauss curvature have Saddle points qualitatively, as against the ellipsoidal points. (synclastic/anticlastic etc.)
The real parts of $ ( x+ i y)^2 = ( x^2 -y^2+  i \,2 x y )\quad \, ( z= 2 xy ;\, z= x^2 -y^2 ) $ are intrinsically same, one can be obtained from the other by rotation through$ 45^0 $ about z-axis.Are ruled surfaces. The surface is called a hyperbolic paraboloid due to cross sections as parabolas/hyperbolas. Hypar is a term used in Civil engineering application.
AFIK there is no standard saddle implying a fixed geometrical parametrization and there perhaps need not be. It is just contrasted from convex bulbous surface geometries.
