They don't have to, they're registered as a Coast Station and coast stations are permitted to identify as their registered name in ITU treaties.
On CW they're required to use call sign.
On RT, they can use: USCG COMMSTAT KODIAK or they can use call sign. Either one.
In UK, EJM Malin Head Radio used EJM on WT, but on 2182 and other WT frequencies, they used "Malin Head Radio".
Ships however did not have this exemption, they were required to use call sign and they could give the name of the vessel also. But things changed - see below!
With recent changes allowing domestic small vessels to use VHF radiotelephone without a ship station license, they had NO call sign. If these vessels went to another country, they'd have to get a station license, and a Marine Radio Operator permit to comply with our treaties.
You may identify your ship station over the air using your FCC-issued
call sign, maritime mobile service identity (MMSI), the state
registration number or official number of your ship, or the name of your
ship.
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DR