The Trick is a substitution, f[x, y] -> g[x, y]/(x - y), in order to eliminate the (x - y) term from the ODE.
("Of course this is not my trick")

(Unevaluated[(x - y)*D[f[x, y], x, y] - D[f[x, y], x] + D[f[x, y], y]] /. f[x, y] -> g[x, y]/(x - y)) // Simplify
(* Derivative[1, 1][g][x, y] *)
DSolve[Derivative[1, 1][g][x, y] == 0, g[x, y], {x, y}]
(* {{g[x, y] -> C[1][x] + C[2][y]}} *)

Then: $$f(x,y)=\frac{c_1(x)+c_2(y)}{x-y}$$ Yes Maple is powerfull software.

Since you provide no boundary conditions, a similarity solution with eta=x/y can be found in Mathematica:

eq = HoldForm[(x - y)*D[f[x, y], x, y] - D[f[x, y], x] + D[f[x, y], y]]


eqt = eq /. f[x, y] -> f[eta] /. eta -> x/y


ReleaseHold[eqt]

-((x Derivative[1][f][x/y])/y^2) - 
 Derivative[1][f][x/
  y]/y + (x - y) (-(Derivative[1][f][x/y]/y^2) - (
    x (f^\[Prime]\[Prime])[x/y])/y^3)

% /. x -> y eta

-(Derivative[1][f][eta]/y) - (
 eta Derivative[1][f][
   eta])/y + (-y + eta y) (-(Derivative[1][f][eta]/y^2) - (
    eta (f^\[Prime]\[Prime])[eta])/y^2)

% y

y (-(Derivative[1][f][eta]/y) - (eta Derivative[1][f][eta])/
   y + (-y + eta y) (-(Derivative[1][f][eta]/y^2) - (
      eta (f^\[Prime]\[Prime])[eta])/y^2))

Simplify[%]

eta (-2 Derivative[1][f][eta] - (-1 + eta) (f^\[Prime]\[Prime])[eta])

DSolve[% == 0, f, eta]

{{f -> Function[{eta}, -(C[1]/(-1 + eta)) + C[2]]}}