After experimenting with it in the beta channel, then doing a bit of A/B testing, it seems that Google is finally ready to roll out bottom navigation in the YouTube app. The update is a combination of an app update, which happened last week, and a server-side update, which is happening now. Assuming you have the former, the later should be coming to you over the course of the next few hours.
Once you have it, gone will be the top tab navigation for Home, Trending, Subscriptions, Shared and Library. Instead, they are all clearly identified on the bottom of the display in the streaming content app.