The count gives both of them a significant line of credit: money overcomes much suspicion of this pair. Their assumed identities are a weapon, and I do not think the scam they are parties to helps your case.
Well, "their assumed identities are a weapon" is precisely my case, and I don't even argue that G.I. identification is actually that great of a solution.
The con tricks are as old as humanity, even if they take different forms in different eras, but the ground problem is the same: if someone approcaches you and claims to be e.g. an important noble named such and such from the overseas, they could very well be telling truth—or they could be lying, and there is almost no way to tell for certain, even though there are some good heuristics (their wealth is one, as you allude to).
Byt why did people believe the count? IIRC, he had pulled some quite elaborate scheme to get the recognition and respect in the Paris, and he was also introduced there by Albert de Morcerf.