We have uncovered a mold problem on the inside of this room. It is an addition to our old 1947 home. This picture is above the window which wasn't properly sealed, so water was leaking right into it creating black mold on inside. We checked the outside and took this picture because we haven't seen this kind of siding before. Want to also make sure there is no asbestos in the board.
The only way to know if something has asbestos is to have it tested. Period.
Having said that, I've never heard of OSB having asbestos.
You said you have never seen this kind of siding before, but it sure looks like common cheap vinyl siding, probably the most common type of siding today. Perhaps you mean sheathing boards? Regardless, almost certain you will not find asbestos there.
But again, the only way to know is to have it tested. Mail-order asbestos testing is $15. So why not, if it makes you feel better?
"Is this old moldy fiberboard..."
It looks like old moldy particle board to me, which is a bit different from fiberboard. As it applies to repair, I don't think the distinction matters much (but you didn't ask about repairing it).
"Want to... make sure there is no asbestos in the board"
If you can establish the date that the addition was built, you might be able to have some confidence about whether it is ACM (asbestos containing material) by establishing whether particle board with asbestos was manufactured in your area at that time. Otherwise you need to send a sample for lab analysis.
BTW, some simple precautions (dust minimization and capture, PPE, etc.) can minimize exposure during repairs. In some locales work with ACM is highly regulated, so... consider whether you really want to know, or just do the work with precautions in place...
I see no sign of significant mold there.
FYI, you have fiberboard siding, a.k.a. buffalo board. It's not OSB, and it's not particle board. It's a low-cost, structural sheathing in common use in the 1980s and '90s.
Unfortunately you don't have a drainage membrane (house wrap), and you presumably don't have modern window flashing (sill pan, rubber tape). This was a low point in construction where a lot of what you've encountered was the result. The only full and complete fix would be to pull the siding off the wall, install house wrap and flashing tape, and reinstall the siding.
