I believe this flat roof surface is modified bitumen but would like to get confirmation. I noted worn surface with fiberglass showing in areas. Am I correct and with the worn areas is this roof at the end of it’s lifespan? I did note a dried moisture stain on the ceiling below but the leak may of come from the old shingled roof that has been replaced.
Looks more like built-up roll roofing with a failed coating. Whatever it is, it should have been replaced with the new shingles. A good coating may save it for a few years.
You might try something like this
Looks like missing flashing and exposed roof sheathing in pic 4 of 4. That might be the source of the leak/stain.
Looks like someone went after roll roofing with a power washer.