The short and simple answer for Eastern European Jewish immigrants to the United States is that they already had several versions of their names when they were in Europe.
If they were Jewish, they had their religious Hebrew name for the synagogue, their Yiddish name that for friends and their Russian name that they gave for their birth and marriage certificates. Most of those name would have seemed strange and unspellable to Americans.
Since they came here intending to become Americans, they had to change their names so they could blend in, get jobs and meet people and escape prejudices.
Sometimes they changed their first names or surnames after they arrived in the USA as they tried to blend in their new homes. Sometimes a Census enumerator who heard the name orally simply wrote it down incorrectly. Often the person who read the handwritten names and created the searchable index did not read the handwriting correctly and made a guess. In one case in my wife’s family, an indexer saw a stylized capital script “I” and read it as an “S”. The typical soundex index searches failed, but it was still possible to find the correct family.
If you are searching recent Census records or naturalization records you want to look for their Americanized name. If you are looking for their Ellis Island immigration record you have to search on their Yiddish or Hebrew names. You probably know their Americanized names but you will have to find out their original names to find that ship manifest.