Are freebie mobile games supposed to show up in your pc library?

I was trying to figure out if I could view achievements for one of the games and I can't find it in the app, the PC or even the play store. Do I only have the game as long as it's downloaded on my device or is it supposed to be in my library? Basically do I not understand how this works or can I justifiable bug tech support to fix it?

I tried seeing if some had asked this before and I don't think they did.