In the past celebrities would adopt an anglicized name to hide their ethnicity of origin.
Did that practice stop or are there celebrities that go by a name that is different from their birth name?

Celebrities seem to be using their birth names nowadays. Those of them who do change is usually because they have the same name as another established actor. For example Michael J Fox added the middle initial to his name because there was already a Michael Fox. (British actor) In the past the movie studios would change an actors name to be easily remembered or their birth names were too mundane or foreign. That practice has fallen along the wayside. It's nice that an actor can keep his own name especially if he is proud of it or his heritage.