For example: My grandfather is named Athanasios and was assigned the name Tom which has an actual Greek equivalent (Tomas). Athanasios doesn’t remotely resemble “Tom” - but every Athansios that I know calls themselves Tom.
Unfortunately there isn't really an answer here, as your question hits on a very common myth about American immigration, as names actually weren't being "Americanized" by the officials en masse at Ellis Island! /u/edhistory101 covers this nicely in this older answer. Changes happened, but they were generally by those holding the name, not forced upon them.