obtuse
An isosceles trapezoid
Two pairs of adjacent sides are congruent. The angles between the non-congruent sides are congruent.
A General Quadrilateral
Not sure about angels, but if angles will satisfy you, the answer is a rectangle
The quadrilateral described is a rhombus. A rhombus has all four sides of equal length and opposite angles that are congruent, with adjacent angles being supplementary. This means it can have two distinct pairs of congruent angles, satisfying the condition mentioned. Additionally, a rhombus can be considered a special type of parallelogram.
a rectangle
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).
Adjacent angles
An isosceles trapezoid
Two pairs of adjacent sides are congruent. The angles between the non-congruent sides are congruent.
A parallelogram
A kite is a quadrilateral with two pairs of adjacent congruent sides. The angles between the unequal sides are equal.
square
A rectangle has two pairs of congruent sides, but any pair of adjacent sides is non-congruent.
This is a rectangle.
A General Quadrilateral
You have described a rectangle.