In A View from the Bridge, Eddie's use of "name" in "I want my name!…Marco's got my name." (Act 2), is a literary device called a trope, which is a literary technique, meaning it is an optional choice the author makes for specific purposes. The trope, one of many kinds, selected by Arthur Miller is called a metonymy.
The definition of the trope metonymy will help explain the meaning of Eddie's line. A metonymy is a substitution of one simple, common word or phrase for a more elaborate or complicated word or phrase. A metonymy is employed for dramatic effect and for simplification of understanding. Two common examples of metonymy are "the White House" and "the sweat of your brow." Examples of their use are:
"The White House declared it would not back down."
"He lived by the sweat of his brow."
Taken literally, these sentences are meaningless; a house, white or otherwise, can not back down, up or sideways, and no one has ever earned a living by selling their...perspiration. In actuality, "White House" substitutes for the power of the U.S. presidency and the decision of the sitting President. The substitution of the seat of power--the White House--summons up recollection of--or at least a feeling of--the Constitutional powers and the distinguished history of the presidency: It's not just one man talking; it's one man invested with the power of the people through the Constitution. This metonymy is dramatic and invokes a powerful imagery.
The same explanation holds true for "sweat of the brow," which substitutes for hours and years--maybe decades--of hard manual labor from which earnings are gained with much struggle. This too is a substitution that is dramatic and carries a powerful imagery with it.
The metonymy that Miller has Eddie speak, "my name," substitutes for the concept of an honest life lived with resppectability and high principles. A variation is "my good name." Maybe Miller had Eddie leave out "good" because Eddie wasn't convinced he had a "good name," meaning a good and honorable life lived genuinely year after year and hardship after hardship.
Marco spits at Eddie--a huge insult--and tells the whole neighborhood that Eddie called Immigration. This destroyed Eddie's "good name," his reputation for being an honorable, respectable man. Eddie says, "I want my name!" because he wants Marco to make the situation right by taking back his accusation and thereby removing the shame and enmity his accusation brought on Eddie--his neighbors all turned their backs on him and won't talk to him.
Eddie says "Marco's got my name." because it lies with Marco, and Marco alone, to clear the cloud of guilt and betrayal from Eddie's reputation--Marco's got his good name in his power: he can clear Eddie's name by saying that, no, he made a mistake; it wasn't Eddie who called.