On his first solo album Mainly Norfolk, Peter Bellamy sang a song called The 14th Of February, learned from Bob Bayfield of Wells-next-the-Sea, with interpolated verses from a Sam Larner version, which began

On the 14th of February we sailed from the land
On the bold Princess Royal bound for Newfoundland

which I think is a version of Bob Roberts' sea-battle song Bubblyrat quotes above, or at any event concerns a battle, in this case against a pirate. The air is not related to the Princess·Royal/Miss·MacDermott tune which is the subject of this thread, but is a completely independent work.

It does indeed show that Princess Royal could be the name of a ship as well as of a person; but the well-known dance tune we are concerned here with I have always taken to have been originally dedicated, like Carolan's reworking, to a particular lady ~ in this case a royal one.