On what device model is the Apple TV app, where you search for this? Is it a smart TV or a TV-connected streaming device? What brand and model? Do you know its manufacturing year? These details may be important.
Some device models get an edition of the Apple TV app without Library or Store (presumably to not compete with the device manufacturer’s own movie Store). If that is the case for you, then your model isn’t supported for Store purchases. Then a solution could be to attach a streaming device that does support the full Apple TV app that does have a user Library and Store (or buy a newer TV that does have this same full support).
Fully supported external streaming devices are: Apple TV box, Roku, Fire TV, Xbox (One, One S, Series X, Series S), PlayStation (4, 5), Nvidia Shield (Android TV 8+), ‘Chromecast with Google TV’ dongle.
Fully supported smart TVs include relatively modern LG (~2018 and later), Samsung (~2018 and later), Vizio (~2016 and later), Sony (Android TV 8+/Google TV), Philips (Android TV 8+), HiSense (VIDAA, Android TV 8+), TCL (Android TV 8+), Panasonic (Android TV 8+), Roku (Roku OS), Fire TV (Fire OS).