Difficult to say, but it could be tretex. In that case, it doesn't hold much. It's the same material as old traditional pinboards, which you can easily push thumbtacks into. If you press against the material at the edge, you should be able to feel if it's soft.
I have successfully mounted curtain rods to a tretex wall by using drywall anchors of this type:
https://www.bauhaus.se/gipsankare-stabilit-22mm-5-pack
They are also available in various versions at Biltema.
However, these won't work if the board is too hard. In that case, a regular screw will probably suffice. When mounting on slightly uncertain fiberboard, like masonite but thicker, I have used drywall screws. These are screws with slightly coarser threads than regular wood screws (but not as coarse as the drywall anchor). They hold more securely in soft or semi-soft materials.
I have successfully mounted curtain rods to a tretex wall by using drywall anchors of this type:
https://www.bauhaus.se/gipsankare-stabilit-22mm-5-pack
They are also available in various versions at Biltema.
However, these won't work if the board is too hard. In that case, a regular screw will probably suffice. When mounting on slightly uncertain fiberboard, like masonite but thicker, I have used drywall screws. These are screws with slightly coarser threads than regular wood screws (but not as coarse as the drywall anchor). They hold more securely in soft or semi-soft materials.
I agree that it resembles treetex. Scrape a little on the edge of the hole to see if it's brown and tends to layer like pressed cardboard; if so, it's treetex. The next question is whether there's a void behind the board or if there is wood there. Many (most) houses from the 50s and 60s have raw planks against the attic or floor joists between floors, which were then covered with masonite or treetex. If you have raw planks behind the boards, you can screw almost anything there without any problem.
