The isohedra have identical faces. They have symmetries acting transitively on their faces -- any face can be mapped to any other face to give the same figure.
There are also polyhedra where all faces are the same, but the faces are not transitive. For example, take an antiprism and make caps with the same triangles.
I just found that this net seems to work, with all faces identical. The long edges all have length 1, with angles of 60 and 90 degrees.
Have polyhedra like this been explored? Is there a name for non-isohedra where all faces are the same?
I just posted a question about these polyhedra, which are monohedral but not isohedral. I've included your shape in the question, which seems to have been unknown at least prior to 1996. I've included every example I know of there, though I very much doubt it encompasses all known non-isohedral convex monohedra.