Sounds a little bit like Speed Dating.
Some sites have this type of video feature.
The only one I can think of now is POF which has a game called NextDate in the Live! section of their app.
It allows you to have short video dates with different people.
Here I found a post about it here: POF Launches New Event to Help Users Virtually Date
They also have another one called Blind Date. The hook with this one is that you see a blurred version of the person at first which slowly clears up as the date progresses.
Plenty of Fish Adds a New Blind Date Game to its Platform