An update on this one. Contrary to my previous understanding, I've discovered that the Mini Player actually remembers its last position as long as the X button (close button) on the player is positioned above the taskbar line. It's probably by designed to ensure that the Mini Player remains accessible when it is placed in an area where the cursor cannot easily reach it.
In my case, I use "TranslucentTb" to make the taskbar translucent and I position the Mini Player above it. Even though this makes the player unresponsive after a while (I can't click on it or generally interact with it when the Mini Player goes in the background) I can still see the album art, song title, artist, and duration of each song.
Is it possible to have an option that allows the Mini Player to remember its last position, regardless of where it is placed on the screen?