The list of games you “own” is populated by your purchases. You are welcome to add it to a collection to have it appear in your library.
Also, you can pay for the game to have it show up in the list of things you own.
Viewing post in How do I add this game to my library?
I have 8 free games in my library listed under "Things you own". There's a "Download" button under each of the game tiles. "The Things We Lost In The Flood" doesn't appear under "Things you own". I added it to my collection, but there's no "Download" button. How do I get the game to register as owned?
Also, why do you say "own" with quotation marks? This is like Steam or GOG, right?