You can eventually trademark once you get big enough. As with all things law it’s a bit tricky. However, the default is that geographic locations aren’t trademarkable.
For further reading on when you can trademark.
https://www.yospinlaw.com/2016/06/15/trademark-on-a-geographical-location
usr does mean user. It was the place for user managed stuff originally. The home directory used to be a sub directory of the usr directory.
The meaning and purpose of unix directories has very organically evolved. Heck, it’s still evolving. For example, the new .config directory in the home directory.