An Immortal object badge is shown in direct relation to Kirito's actions. What are these badges referring to? Was Kirito immortal?
You're likely thinking of the Immortal Objects which Kirito (and other characters) have been known to strike at.
In Sword Art Online, physical surroundings such as walls, buildings, monuments, and major features are considered "Immortal Objects". An error beep and a notification will sound if immortal physical objects are struck, even if by accident.
This doesn't change the state of the player's mortality - they are very much mortal - it only informs of the state of the object they want to interact with. If it's an Immortal Object, it can't take any damage, so striking it (regardless of intent) will not cause harm to it.