Yes, it's possible.
There is a very easy way, and then a more difficult way which will be required for the layout you show.
For the easy way, you would need to make top categories of "Categories" and "Manufacturers", with all of the products organized in categories under each of them (and the ordinary non-expanded display). When you click on Categories or Manufacturers you would see the top categories under that main division.
But that won't allow the display you want, with categories completely expanded, or the two buttons side by side at the top.
Probably what you want is to display either the categories box or the manufacturers box depending on which button is clicked. It can likely be done with javascript, but I don't know about doing that.
I presume you would want the display to stay the same from page to page until the other button is clicked, so you would probably need to create a session variable or a global variable. That is getting out of my range of experience.
You would put a test at the beginning of tpl_categories.php and the manufacturers box file, so the one set to "on" would execute and the other one would not. I would have to look up more detail and don't have time right now. Others here can give more detailed advice.