Your friend isn't the first musician to share a name with an automobile. Consider REO Speedwagon, The Valiants, The Road Runners, The Avantis, The Mustangs, The Rivieras, The Ferraris, The Lincolns, and our musical favorites, Galaxie 500. The difference is that in your friend's case, he was using the name before the car existed.
Is there a claim against the car company? Even though your friend may have invented the name and used it first, we don't believe he has a trademark claim against the car company. That's because trademark disputes are resolved using a likelihood of confusion standard. Are purchasers of the Korean car likely to be confused as to whether it originated with your friend's musical project? Are consumers of your friend's records likely to be confused into thinking that the source for the music is the Korean car company? Because the categories of goods are so different (and don't compete with each other), and because purchasers of cars and music are discerning enough to distinguish between the two, we think that consumer confusion is unlikely.
Money from the car company? Because confusion is unlikely, the car company need not pay your friend for the right to use the name. The example you mentioned -- one band paying off another band for the right to use the name -- differs from your friend's situation because consumer confusion is much more likely when two bands have the same name. That doesn't mean car companies can name their vehicles, The Beyonce, or the Beatles. Using a famous band's name implies endorsement and violates unfair business and right of publicity laws.
Should the name be registered? Federal trademark registration offers benefits but is not mandatory. If your friend's right to the name is challenged, the most important factor will be evidence of where and when he used it in commerce -- for example, advertisements, reviews, discography, etc.