Ellis Island clerks were not allowed to arbitrarily change an immigrant's name. The name had to be verified by several sources so that an identity could not be mistaken. The casual name change is an urban legend, spread partly due to a scene in The Godfather Part II (1974).
The gang's train tickets all have the same seat number.
The gang have apparently finished eating at the restaurant as they've paid their check, yet there's a lone milkshake (with the addition of not just cream, but a cherry, too) just sitting on the table the entire time the gang are talking. They even walk away from it, without Scooby or Shaggy taking a sip, which seems highly improbable that they would leave it.
The train's schedule board leaves out Transylvania, their destination.