“While Britain is an island that retains a monarchy,”
Unlike the US where the President is both the Head of State as well as the Head of Government, in Britain the PM is the Head of Government while the King/Queen is the Head of State. There are no major countries in Europe where these functions reside solely in a monarch.
Yes but even in its symbolic sense: the British monarchy distinguishes British identity, its sense of sovereignty, and its relationship towards Europe. (When comparing to France/Germany in particular.)
And as stated before, it has a closer bond to the U.S. historically (for obvious reasons) and presides over a thriving Commonwealth of nations.