While playing Persona 5 (finally) last night, I came across a funny scene that I wanted to show a friend. However, when I hit the Share button, the "Save Screenshot" and "Save Video Clip" options were disabled. I am completely sure that I have enough space on my system to save screenshots or video clips, so what's going on?
In order to prevent spoilers, ATLUS blocked the PS4's native share functionality for the entirety of the game. This includes saving screenshots and videos. Thus, this isn't an issue with your PS4, it's a restriction placed on the game itself.
For reference, see this article:
Atlus confirmed to Polygon that because Persona 5 is such a story-based game, the company made the decision to prohibit players from sharing content.
“This being a Japanese title with solely a single-playthrough story means Japan is very wary about it,” an Atlus representative said. “Sharing is currently blocked through the native PS4 UI.”
Note: as mentioned by @MadMAxJr in the comments, this includes screenshots typically taken when earning trophies. Thus, there will not be any screenshots taken when earning trophies in this game.