Can you ID these storage bins?
According to Liam Nilsen from The LEGO Idea Studio, he believes the storage bins may have been produced in-house by LEGO and are not commercially available. 



The online portfolio for GXN (the designers of the studio) mentions:


  Bespoke elements are accompanied by Open Desk furniture pieces, all
  fabricated by local makers and assembled on site.


I couldn't find the drawers on the Open Desk furniture site but it seems the parts containers are designed to fit snugly into the drawers.