In a post made on Google’s support page, it reads, “Two years ago, we launched a feature to enable you to share videos via direct messages on YouTube. Since then, we’ve also focused on public conversations with updates to comments, posts, and stories. We’re constantly reevaluating our priorities and have decided to discontinue YouTube’s native direct messaging feature while we focus on improving public conversations.”
This feature is expected to be removed from the web version of YouTube as well as the mobile apps on iOS and Android. It is a bit of a pity to see the feature being removed as it was rather handy, but then again, there are probably too many messaging apps around at the moment where users might want to avoid confusion by just sticking to what they know.
The feature will be removed on the 18th of September so you still have about less than a month to keep using it if you want to.