foo /: MakeBoxes[foo[x_, y_], StandardForm] := ToBoxes @ Graphics[{
    Point@{x, y}
  }, Frame -> True, PlotRange -> {{-2, 2}, {-2, 2}}
]

foobar[foo[x_, y_]] := (Print["Matched"]; x + y)

foo2[x_, y_] := Interpretation[
    Graphics[{
        Point@{x, y}
      }, Frame -> True, PlotRange -> {{-2, 2}, {-2, 2}}
    ],
    foo2[x, y]
]

foobar2[foo2[x_, y_]] := (Print["Matched"]; x + y)