A possible solution can be reached by building
two different 2-row pattern, which don't have a common tetromino, and inside themselves do not have touching tetrominoes of the same type.
A pair of possible patterns of this kind is:
which both can be repeated infinite times to 'fill' a $2\times \mathbb Z$ part of the plane. Placing them below each other in an interchanging order you get a tessellation of the plane. Any of the 2-row elements can be translated horizontally, so this already gives infinite solutions.
For clarification, here is a picture.
In this one, I enhanced the first 2-row pattern with the square tetromino, so now each type is used in the tessellation. Also this version has a $4\times8$ tiles repetition. By putting this kind of blocks next to each other (both horizontally and vertically), you get a tiling which suits the criteria.