At Budapest’s Keleti Train Station last week, Mahmoud, a Syrian from Aleppo, looked around the underground concourse packed with new arrivals like himself. Judging from their accents and dialects, he reckoned that little more than 10% of them were Syrian. But he saw many more passing themselves off as Syrians.

Indeed, during his journey through Greece and the Balkans on his way to Hungary, “I found a bunch of Iraqis buying fake Syrian passports,” said Mahmoud, adding that now Syrians “are worried that their passports are being...