Infinitely many.
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2 pairs, the sides opposite each other are congruent to each other
A parallelogram has two pairs of congruent sides, and the opposite sides are congruent.
2 pairs
This is a parallelogram. The first requirement is 2 pairs of congruent sides where the congruent sides are not adjacent. This is like a rectangle (excluding a square) that has two pairs of congruent sides where the congruent sides are not adjacent. But the angles are not all congruent (as set in the question) which pushes the shape into the "next less regular" shape, the parallelogram. The angles will not all be congruent, but it will have 2 pairs of congruent angles. There is no way to avoid the 2 pairs of congruent angles because of the requirement that the shape must have 2 pairs of congruent sides (the first requirement).
Six. If the sides are labelled a, b, c and d then the congruent pairs are: ab, ac, ad, bc, bd and cd
2 pairs, the sides opposite each other are congruent to each other
A parallelogram has two pairs of congruent sides, and the opposite sides are congruent.
In a rhombus, all of the sides are congruent, so 4.
Opposite sides of a parallelogram are congruent and so it has 2 pairs of congruent sides.
2 pairs
This is a parallelogram. The first requirement is 2 pairs of congruent sides where the congruent sides are not adjacent. This is like a rectangle (excluding a square) that has two pairs of congruent sides where the congruent sides are not adjacent. But the angles are not all congruent (as set in the question) which pushes the shape into the "next less regular" shape, the parallelogram. The angles will not all be congruent, but it will have 2 pairs of congruent angles. There is no way to avoid the 2 pairs of congruent angles because of the requirement that the shape must have 2 pairs of congruent sides (the first requirement).
Six. If the sides are labelled a, b, c and d then the congruent pairs are: ab, ac, ad, bc, bd and cd
A quadrilateral with both pairs of opposite sides congruent is a parallelogram.
Diamond
A rectangle has 4 congruent angles. 90 degrees. It is also a parallelogram, 2 pairs of congruent sides.
One pair
This shape would be a square.