Two ps4’s under same account?

Ok so I have a ps4, connected to a RAID drive to save my digital downloads to. I don’t buy disks hence the RAID. Anyway, planning on getting another ps4 for my daughter so she can play terraria, dreams, little big planet, among others. All of these are digital downloads on my machine. If I get her a PS4, can she download the games and play while I am? Of course I’ll be playing a diff game, I don’t do multiplayer (hence why I mentioned the game dreams, it’s all user based content, and online). Of course she won’t have a 300+ full game library, just what she wants that I already bought. 

So, is that possible? I’m not sure bc she would be logged in under my user name to download, and be “online” same time as me. For example if my internet drops, the PS4 nags me about it and verifying that I own the license to play the game stored on my array of hard drives. Yeah as if that makes sense... why does Sony gotta constantly check if I bought the game, when duh it’s on my account history, online or offline?!