Polygons can have widely arbitrary edge lengths and angles. The correctness of a tetrad has to be proved.
A polyomino can have only angles which are multiples of 90° and the length of an edge is an integer.
It can easily be seen whether a polyomino tetrad is correct or not.
Here I describe the way I found my first tetrads made of congruent polyominoes in 1971.
I started with a graph that occurs in the answer of the last question.
Use vertical lines instead of oblique lines.
Move the upper polyomino to the left. Then it touches the green polyomino.
Add a rectangle to the upper polyomino. Then it touches the blue polyomino.
For the next steps it is necessary that the added rectangle has the same length as the bottom rectangle of an upright polyomino.
This was already taken in account before.
The rectangle also has to be added to the left hand sides of the other three polyominoes.
The added green rectangle cuts off a part of the yellow polyomino.
And the yellow rectangle cuts off a part of the yellow polyomino.
The same areas have to be cut from the red and green polyominoes.
The yellow polyomino is the mirror image of the other polyominoes.
In order to avoid this, we move the red polyomino, together with a part of the yellow polyomino, one step to the left.
This was my first tetrad made of polyominoes. Each polyomino is of order 20.
It would be nice to reduce the order of the polyominoes. To achieve order 12 was not very difficult.
The following pictures do not need comments.