I absolutely love Amarok, and I want to thank the devs for the great work they've done with this software.

I've encountered a problem the way Amarok handles albums; it's not a critical problem and it's easy enough to work around, but it seems like there should be a way to fix it.

I have Neil Young's Greatest Hits album, which is a little unusual in that it has several tracks by Neil alone, and some by Crosby, Stills, Nash, and Young, and some with Neil Young and Crazy Horse. This puts it in the 'Various Artists' heading in the "artist/album" view. Fine, except that I also have a Greatest Hits album by Al Green, and Amarok likes to think they are all part of the same album. It doesn't show up in 'Various Artists' (it's under 'Al Green'), but the two albums share a cover and when I tried to add the whole album when the context feature listed albums by 'Al Green', I got it mixed in with Neil's stuff.

I don't know what the solution to this would be besides some sort of album ID that isn't based on title. Is there already a way to handle this in Amarok?

This IS a known issue, and has been for ever. The handling for "various artist" albums, as well as "album artist" is being reworked for Amarok2, and that should help at least with where the albums show up.
In the meantime (and this is how I worked around it in Amarok 1.4) tagging creatively is the best way to deal with the issue. For example, Even though the album's official name is "Greatest Hits", I would use "Al Green - Greatest Hits" (for example).
That said, I never had an issue with album covers even without the above tagging - although you don't specify which version of Amarok you're using. I've only extensively used 1.4, haven't played a lot with 2.0.

I never had this problem in Amarok 1.3 or 1.4

I have the same collection and in 2.0 I had to re-tag all the compliations and some of them refuse to be identified as various artists.