Look at the example 3 in the following picture and also its orbits.
It is from the book: Algebraic Geometry IV: Linear Algebraic Groups Invariant Theory by A.N. Parshin，I.R. Shafarevich. The author treat it as a counterexample which categorical quotient does not exist.
But in another book, Actions and Invariants of Algebraic Groups, Second Edition,by Walter Ricardo Ferrer Santos, Alvaro Rittatore, there is a similar example which has categorical quotient, but no orbit space or geometric quotient.
Does this action have categorical quotient?
I think yes by intuition, but can not show the exact reason.