You don’t open a new Audacity page (new window), that won’t work.
You create a second track in the same Audacity page (window) as the backing track, see … https://youtu.be/1b-39uw0D1s?t=2m10s
When I play back MP3 song in Audacity and push the “record” button, it does nothing as though the record button is not active (however it’s not greyed out).
While playing back a MP3 song in Audacity, when I press “record” button, I do not have any mic attached. Is this the reason nothing happened even when “record” is pressed?
Are you trying to make a copy of the MP3 within the project or across projects? If so, then there’s no need to record at all; just use Edit → Duplicate (or Ctrl+D), or Edit → Copy (or Ctrl+C) followed by Edit → Paste (or Ctrl+V) in the desired window.
The record button will be greyed out while Audacity is playing. If you want to record, stop playback first, then click the record button and Audacity will begin recording.
What exactly are you trying to do? If you just want a copy of an MP3 file, you don’t need to use Audacity or any other software - just right click on the file in your computer’s file explorer and select “Copy”, then go to the folder where you want the duplicate and right click > Paste.
It doesn’t make a lot of sense to me but I do notice that your English language skills are very good. If you find Audacity to be a useful program, perhaps you would be interested in helping to translate it into Japanese, as that is one of the languages where the translation is very incomplete. There’s some information for translation volunteers here: https://www.audacityteam.org/community/translators/