Thor, Son of Odin, is still Thor (the person).
Thor is no longer worthy (as depicted in a recent issue of New Avengers) and loses the hammer Mjolnir to Female Thor (who presumably also has a real-name, but I'm not sure what).
Thor, 3000 year old Son of Odin still exists, but epithet of Hammer-wielding Avenger changed hands. It's just weird because unlike Tony Stark or Steve Rogers or Bruce Wayne his superhero identity is the same name as his actual name.
The superhero Thor is just whomever currently wields the hammer.