None.
The only common property of quadrilaterals is that they are plane figures that are bounded by four straight sides, and that they have four vertices. So far all quadrilaterals are candidates for the answer. But then with each quadrilateral you get some unique property that is not shared by others so NO quadrilateral has all te properties of all the other quadrilaterals.
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All squares are quadrilaterals. All quadrilaterals are not squares. There is no square that is not a quadrilateral.
All quadrilaterals are polygons, but not all polygons are quadrilaterals.
A rectangle is a quadrilateral. As such, it shares the distinguishing characteristic of quadrilaterals with all other quadrilaterals, and has four (4) sides.
No, all quadrilaterals are not parallelograms.
Quadrilateral is the broader term. All kites are quadrilaterals, but not all quadrilaterals are kites.