The hyperplane is the zero set of an exact rational score. Here the displayed w and b make a simple line, and the sign of the score gives the side.
The separating hyperplane
Use w=(1/2,1/2), b=-1. Then w·x+b=0 is the line x+y=2.
w⋅x+b=0⟺x+y=2
Why the sign is enough
The sign of w·x+b puts a point on one side or the other. This is an exact rational sign check for the shown coordinates.
sign(w⋅x+b) is exact here
Summary
The line x+y=2 is not fitted by a rendered optimizer. It is the displayed separator that later checks will verify.