I'm certainly not an expert in boating, but I can give a possible answer about the meaning and the origin of the term doppiare in nautical language.
And about its connection with the term doppio.
Nautical terminology is very complicated and there are dictionaries on this subject.
The following is a Dizionario del mare (Dictionary of the Sea), of 1932, and the entry doppiare can shed light on our problem:
I put the image of the entry doppiare below:
Doppiare termine proprio della marina. Significa passare a breve distanza, descrivendo un breve giro, dall’una all’altra parte di un capo, di una punta, di un'isola. Quando si passa in linea retta si dice: montare.
The dictionary describes two ways of passing through a cape, doppiare and montare.
Doppiare un capo means passing a short distance from a cape (or an island), describing a turn. Instead, when the cape is passed along a straight line the term is montare.
Therefore, but this is my conjecture, when a boat doppia a cape, it goes, in a sense, twice along the cape: it goes along the first side, then it passes the point of the cape and then goes again along the other side of the cape.
This seems to me not very different from the English term 'round', which also suggests a curved movement, not a straight line.
And the origin from the Spanish term doblar, mentioned by Denis Nardin in the comments, seems to confirm this connection between curvare, girare and raddoppiare, both present in the Spanish term doblar.