I believe it is only currency and possibly stamps.
Passports, green cards, etc I think would fall under customs. If it is a cross border counterfeiting ring I think the FBI would be involved.
Its all financial instruments.
Also, CBP and ICE are also in DHS, so between those and USSS, it covers everything but private documents (college diplomas, unless those are used to prove residency), ie everything but private fraud. Making a fake license for a citizen (ie a minor to buy booze) is a state LEO responsibility...doing so across state lines, FBI, doing so for an illegal alien, DHS.
So where is the argument? If there were financial instruments or documents used for immigration/residency confirmation, it would fall under DHS.