I have a solution which works, but don't know why.
Change the date of the device (Settings-> Date and Time) to an year greater than 2050. (uncheck set date and time automatically).
Then delete the files from SD card, copy to your computer.
Restart your phone.
Copy the files back to SD card.
Open Music App. The files will refresh and should work flawlessly!
Restart your phone!
PS: I think that this is caused due to the fact that the cache files are stored for a very long time on Windows phones, especially for the system apps. Therefore, if you have made major changes recently, it may not update the cache and hence the files conflict. Increasing to a very advance date, forces the cache to refresh due to the date difference!