Tableau can do everything seen on the attached image and more, except for doing 'Tetris' on the blocks and compacting them nicely together. Each block and group of blocks will be allocated their own compartments, with white space between them.
It may be possible to use Tableau's polygons feature to do the 'Tetris' look, but if you ask me it is more trouble than it's worth.
Do you have an example of a workbook where you created such a Heat Map? I would be very interested to see how Tableau can do it.