a lot of games do this
you can usually only play like a "demo" of the game when it first installs.
it looks as if you have completed the full download and installed the full game because nothing shows up in notifications, but the fact is you have usually only installed 10% of the game at this point.
to see how much of the game has really been downloaded/installed you have to go to the main menu, highlight the game and select options and go to information, this is when you realise the game still has about 15 hours worth of downloading still to do.
hopefully sony bring out an update so it still shows in notifications that it's still downloading, because the current method IMO is stupid as I ended up re-installing a game multiple times as i thought the download file was being corrupted as I only had access to the demo teams and nothing else, no multiplayer, etc.
It does work but you just cannot play very much of the game making it pointless.
It's best to once it has installed checked the info and then stick the console in standby mode overnight.
if you have the console in standby mode iirc games don't install until you turn it on. so you could have a 40GB download the demo part of it could be 4 GB, you could leave the console on for 10 years and it will only download the first 4GB and wait for you to turn the console on to install it before then downloading the other 36GB in the background completely unknown to you because it shows as installed now in notifications.
so it's best to turn it on after a few hours to install the "demo" then put it back into standby.