Is there a foam that hardens like concrete?

Yes, there is foam that hardens like concrete, commonly called concrete foam, foamed concrete, or rigid polyurethane foam that cures to hard solid similar to concrete. Examples include closed cell spray foam that hardens rigidly after application, providing structural support and insulation, though not as hard as concrete. More directly, there is foam concrete which is concrete mixed with foaming agent to create lightweight concrete that hardens like concrete but with air bubbles making it lighter and insulating. There is also expanding structural foam used in construction that hardens to rock-like hardness for filling voids and supporting structures. Epoxy foam and phenolic foam can harden to very rigid, hard solids used in marine and aerospace. For DIY, there is hard expanding foam like Great Stuff Big Gap Filler that hardens rigid, and two-part rigid polyurethane foam used for sculpting and structural support that hardens like wood or concrete. So yes, several foams harden to concrete-like rigidity, though true concrete strength requires cement-based foamed concrete.