The reason there's not a lot of fan-made machinimas/GMVs of FFXIV around is because they generally violate the game's Material License by their very nature (found here - specifically you can't show footage from the game and replace it's music/sound with something else.

Recording function wise, the PS4 has an in-built function for recording in-game footage, but from what I understand you can only record up to half an hour of footage at a time. There are third party programs around for recording game footage as well, like FRAPs and Bandicam, but they're usually not freeware and require you to pay for a license to upgrade them to the full version.

As much as an official 'model/animation viewer/editor' would be an interesting tool to have, I don't see SE adding any such feature to the game. Watching cutscenes through the Unending Journey and directing friends and strangers to perform certain emotes on cue is about the closest you're going to get I'm sorry to say.