I have Exclusive Mode in Amazon Music on my 2018 MacBook Pro running latest Ventura as well as late 2009 iMac running High Sierra.
Tip for Windows: For Mac and Windows, you can download the app through Amazon's website. However, for Windows, you can also download the app through their Microsoft Store. The apps look identical, however, the version downloaded from Microsoft Store also has additional control and removes volume option. I have the app from Microsoft Store installed in my Windows NUC.
I will also say that yesterday there was an update that bricked the app on my Mac. It wouldn't reload. Many others had this issue yesterday as well. I had to re download from Amazon's website which is slightly older but stable. Maybe try doing that and compare the version number of what you're currently running.
Also, I use an external DAC, and exclusive option is only offered for that and not the regular built in card when it is presented as "Default". However, even the built in card shows up as 2 options. One that is listed as default and once again without the Default. Pick the lower option that is not default and you will be able to select Exclusive Mode.
The bottom one you can toggle and the Exclusive Mode switch becomes blue from grey.