Apple TV Library Not showing
I have an LG OLED55C7P-U, which is on the compatible list, but I do not get to see my library on Apple TV+
OLED55C7P-U,
Select relatively new LG TVs would have the full Apple TV app. I think from 2018 or later. Slightly older or not as high end models (2016 and later) get the Apple TV+ app, specialized for the Apple TV+ streaming service, but unfortunately without Store or Library (iTunes Store access), and without other third party in-app channels.
This support document has more details on the differences:
Get the Apple TV app or Apple TV+ app on your smart TV or streaming device - Apple Support
If you want access to the Store and Library on this model TV, then consider connecting an external streaming device that supports the (full) Apple TV app.