Series Purchased in Apple TV can’t be viewed in Apple TV.

if it was me then I would make sure the same appleID was set on the appletv box

Switch the active user profile on Apple TV 4K - Apple Support

How do I switch Apple ID on Apple TV? - Apple Community

everything I ever purchased with my appleID show up in Library on my appletv box, my iphone, ipad, appletv app on computers and the likes (use the same appleID on everything of cause)

Just my iPhone

Alright. Then purchases should be immediately visible there. Did you get a confirmation of the purchase? Like an email? Else try finding it in your purchase history for confirmation. If not found, or listed there but not in your library, then please consider contacting the Apple Media Services staff for further assistance.

(not sure why it matters though)

If it was on a different device, then the troubleshooting should also focus on that device (smart TV, streaming device, computer, etc.), and any deviations in Apple TV app edition there. E.g., a few select devices with the Apple TV app don’t even have a library, i.e. not suitable for watching Store purchases.

… now that I have made my purchase, the only option appears to be to open episodes in Amazon Prime. The show also does not show up in “Purchases” under “Library” in the Apple TV app.

Besides the iPhone, on what other device did you intend to watch in the Apple TV app?