Factor[
  eqn /. {x0 -> Sqrt[u] a, y0 -> Sqrt[w] b}
  ] /. ({u -> x0^2/a^2, w -> y0^2/b^2})
(*  (4 a^2 b^6 (-1 + x0^2/a^2 - y0^2/b^2) (x0^2/a^2 - y0^2/b^2))/y0^2  *)

Also:

sub = {x0 == a Sqrt[u], y0 == b Sqrt[w]};
Factor[
  eqn /. First@Solve[sub, {x0, y0}]
  ] /. First@Normal@Quiet@Solve[sub, {u, w}]

Here is a way that uses FactorList:

eqn = -4 a^2 b^2 (-b^2 +
     (b^4 x0^2)/(a^2 y0^2) -
     (-((b^2 x0^2)/(a^2 y0)) +
        y0)^2);
expr = (x0^2/a^2 - y0^2/b^2);
FactorList[eqn/expr]
% /. {c_Plus, 1} :>
  Splice@{{a^2 b^2, 1},
    {Expand[c/(a^2 b^2)], 1}}
MapApply[Power, %]
expr*Times @@ %