For many artists with the same name, Last.fm could maintain multiple pages. One for each.
Now, how to differentiate?
This will take effort from the community, but I was thinking that songs could be categorized under a disambiguation page until a bunch of people confirm which artist they're by.
For example, I listen to a song tagged Rain. Let's say this song hasn't been categorized yet. Then I could click it in my profile to go to a page labeled Rain (disambiguation). Then I would vote for which Rain said song is by. Let's say I pick 비. Well if a bunch of people agree, then in the future it will scrobble it under 비. For everyone.
It's probably not perfect, so please comment with your thoughts. And some way to suggest this to Last.fm staff. =P
And comment with other artists of this nature if you'd like. I'll add them to the list below.
Other multiple identity artists: