You can't copy and paste directly with Premiere Elements. You can export (share) the audio and then import it to the new project.
What version of Premiere Elements and what operating system?
I believe Peru Bob has misread your question. You seem to be asking about copying and pasting within the same project, not project to project.
From another thread, you seem to be working with Premiere Elements 13, not sure what computer operating system.
Please try the following to accomplish your stated task which is very doable within the one project.
1. You separate the audio from the video. (If you need help on that let us know, but you seem to have done that already.)
2. Right click the unlinked audio clip, select Copy from the pop up menu.
3. Move the Timeline Indicator to the right to a clear space on the Timeline. Then
4. Move your audio copy just generated to the Soundtrack.
Please let us know if that works for you.