If you purchased the boxed packaging of Premiere Elements 13, it should have come with two installation discs for Premiere Elements 13, one for Windows and one for Mac. In this situation, you would use the same serial number whether you are installing Premiere Elements 13 Windows or Premiere Elements 13 Mac.
However, the interpretation of the Adobe Licensing Agreement seems to be that you are allowed the purchased activated serial number on no more than 2 of your computers, and you do not use the 2 at the same time.
Key thought, if you purchase an online download, that download is specific for Premiere Elements 13 Mac OR Premiere Elements 13 Windows. Your purchase is for one OR the other, not like the box packaging where you have a disc for each included in the purchase. It is my understanding that Adobe direct sells only online download, and you would need to go to an authorized reseller to purchase of the box packaged product.
I am still working on the Premiere Elements 13 tryout. So, the above applies to history through version 12 and expected same for 13.
Please let us know if you have questions or need clarification on anything written.