I don't think using Amazon Luna on your PC gives you any particular privacy concerns. It's just a website. If you're worried, as long as you're using a decent privacy-conscious web browser, and/or a separate browser to what youd normally use, IMO you're probably fine. I think Luna needs a Chromium based web browser to work if it helps.

I play Fortnite on my Fedora laptop with my son using my partner's Prime account, and it works pretty well. It pops up at the start to say it's not a supported platform, but then plays with generally no problem.

Oh one thing, you'd need to link your Amazon account and Epic account, so whoever has access to your Amazon account can play your Epic games through Luna. For me that's not an issue but it might be for you if it's your mom's account?