A quadrilateral having both diagonals congruent is a?

If both pairs of opposite sides are parallel: A Rectangle, or a Square.

If exactly one pair of opposite sides are parallel: An Isosceles Trapezoid.

If it does not have parallel sides and one diagonal is the perpendicular bisector of the other: A Kite

It is also possible that it does not have any parallel sides and neither diagonal is the perpendicular bisector of the other: A quadrilateral