I'd have to look at the ship manifest, but based on the marriage record someone found on the other thread, I think Cseh is probably a misreading in this case. It's an understandable error, because Cseh (/cheh/, like "check" with the end cut off; it means Czech) is vastly more common than Czéh (very roughly /tsay/, like "say" but with a 't' added at the start; it means "guild").
And regarding that bit you wrote about Szabó being a familiar name -- well, no wonder: Szabó is a very, very, very common name in Hungarian. Think Smith or Jones level of common. It's generally true in genealogy that you shouldn't base anything on family name alone, and this is especially true for Szabó, Kovács, Tóth, Horváth, and Nagy. (They're the Big 5 of surnames, with 200,000 bearers apiece, in a country of under 10 million.)