The use of aliases or false identities is the subject of many fictional works, including The Importance of Being Earnest. In the play and in other works, characters who attempt to deceive others regarding their identities become entangled in their own lies. They forget what they have told certain people and what they have withheld from others. They don't answer to the name that they have establish with one person because they drift off in thought, forget where they are, etc. Finally, they make appointments or social dates for one identity that start to conflict with their other identity's appointments or relationships.
While I think that most would agree that identity fraud has increased dramatically in today's society versus Oscar Wilde's era, I think that many would also agree that it is a little more difficult to establish, maintain, and pull off alternate identities because of the widespread use of security cameras and the availability of cell phone records (and all the information one can glean from those records).
For a real-life, more modern take on someone creating multiple identities for himself, see Catch Me if You Can with Leonardo DeCaprio which is based on the autobiography of Frank Abagnale who really did carry off portraying himself as a doctor, pilot, etc., and who now serves as a consultant to the FBI and other agencies.