I shoot vertical video when creating content that is specifically aimed for social media (either instagram or Facebook).
I did a series of videos for the San Diego Chargers, and after doing a couple square videos and traditional widescreen 16:9, the square ones got more interaction on Facebook.
I find this to be the same case on Instagram, where a vertical video takes up more real estate in the feed and thus demands more attention. For Instagram feeds though, video can only be 5:4 not 16:9. or you can go square of course.
Several people I follow have gone from shooting standard youtube vlogs to shooting just for Instagram stories, and walk around with vlog setups but have the camera vertical (https://www.instagram.com/jessedriftwood/ probably being the best example). I've shot a couple videos like this while on trips and some commissioned work for music festivals, but it's just kind of a bummer that all the footage can only be used for one specific purpose. It obviously looks awful if you try to share vertical video anywhere on the web.