Yes a single server can usually server multiple users.

When a video is uploaded, it will usually be saved to one of the machines in the uploaded region. The same machine can now serve many many users. This number will depend on several factors like CPU, RAM, SSD, network bandwidth of the machine…

Such servers can now be distributed across the world and hold a copy of it. This helps reduce buffering on the user side as they will be most likely served by a server in a region closest to them. Ex - server in Oregon handles traffic for folks in the west coast of USA, a server in Dublin Ireland will serve users in Western Europe and so on….

(Algorithms to determine how/when this is needs to copied over)

Depending on the demand in each region can now scale up to the more servers to serve the traffic needs in that region.

(Again determined algorithmically)