It's a wood fiber board with a thin asphalt coating often called "buffalo board" or "beaverboard", the trade name of the long defunct Beaver Board Co. It was an inexpensive exterior wall sheathing.

It was also called and known as Celotex sheathing

This might help you out.

do I need to insulate the bays with fiberglass batts or can i just cut the rigid foam board to fit the bays and foam them in place?

putting just foam sheathing inside the joist bays doesn't
give you much insulation value. at best R-6 per inch.

you don't give your location...but in my hot humid climate
once air sealing of floors is done we overfill the joist
bay with fiberglass batt...
this is done to keep insulation in contact
with undersides of floor, as over time the fg batt will settle
& not stay in contact with floor.
use a friction fit batt or the wires to hold the fg batt in place.
then use the foam board & attach it with button cap nails
to the undersides of the fg batt filled floor joists.
from the crawlspace you would see just the foam sheathing
across the length of the floors. caulk the foamboard to the joists, tape the seams.

for 2x6 (5.5" depth) use R-25 to R-30. what you lose in
slight compression will eliminate any issues of condensation between bottom of floor & settled insulation that occurs over time...gravity & all that.

why would you use foil faced foam sheathing? no radiant heat to reflect in a crawlspace.
dow blue board or corning pink board are t&g products that would work better, or so I've found over the years.

as for the black fiberous board..got a picture?
my guess would be baggasse.

best of luck.

Sounds like what a friend of ours (who is a contractor) referred to as "horseboard", which is what we have under our aluminium siding.

Just because the house was built 2 years after they stopped producing product with asbestos doesn't mean you're safe. The only way to be sure is to have a sample tested. They could have used material which was sitting in a warehouse, etc for more than two years.