What's the name of this wooden box around the tree on the sidewalk? Does it have any utility?
The name is "tree seat" or "wrap-around tree bench".
Here are some additional examples of tree seats (wrap-around tree benches):
(photo credit: Moosejaw Woodworks)
(photo credit: STREETLIFE)
The utility in both private and public spaces is multifold: seating, resting, observing birds, sleeping, giving speeches, to name the few.