It can be seen in somme pictures at https://ussconstitutionmuseum.org/2018/01/10/year-in-review-2017/
There is even one where it seems to be towing the warship.

I did not found a name but a nickname, it is called the BEAVER BOOMER and its role seems to be operating the floating barrier at the end of USS CONSTITUTION's wharf and assists when the old ship has to move.

If she don't need sail too far out from the berth, wonder what's her radar for ?

Could be like the one described in this auction. The type is known as a "security tug", "beaver tug" or "barrier boat" at first conceived to move logs in the water and used by the navy "to open or close barriers around naval installations". More information at https://gsaauctions.gov/gsaauctions/aucdsclnk?sl=41QSCI14410001

There are some other pictures of it in the net, the only writing other than "US NAVY" is "NO STEP" on a hatch at the front. Maybe it is the name, the USS NO STEP?
https://www.reddit.com/r/navy/comments/2gi7by/hello_guys_can_anyone_identify_this_tugboat/