This is something that's come up on this sub once or twice and I think it's a misconception based on (as you said) popular culture, fiction and especially video games. During the period I study, it was not at all uncommon for ships' crews to decide engagements by boarding the other ship and fighting hand-to-hand; in fact, this is probably the oldest form of naval warfare inasmuch as heavy artillery wasn't even mounted aboard ships until the very late Middle Ages.

Now that said, captains would absolutely be involved in boarding an enemy ship, and it's more than likely given the sheer volume of engagements that one captain would fight another at some point. Bu the romanticized 1-1 duel seems to have mostly been a trope of fiction, although we have many accounts of men losing heart if their captain was struck down or vice versa.

There is one particularly famous challenge issued by a captain to another ship, in the context of the War of 1812; the full text of this seems not to be easily accessible online and my book that would have it is at work. But the edited version that's often reprinted captures the gist of the message sent from British captain Philip Broke, commander of HMS Shannon, to American captain James Lawrence of USS Chesapeake.

As the Chesapeake appears now ready for sea, I request you will do me the favour to meet the Shannon with her, ship to ship, to try the fortune of our respective flags. The Shannon mounts twenty-four guns upon her broadside and one light boat-gun; 18 pounders upon her maindeck, and 32-pounder carronades upon her quarterdeck and forecastle; and is manned with a complement of 300 men and boys, beside thirty seamen, boys, and passengers, who were taken out of recaptured vessels lately. I entreat you, sir, not to imagine that I am urged by mere personal vanity to the wish of meeting the Chesapeake, or that I depend only upon your personal ambition for your acceding to this invitation. We have both noble motives. You will feel it as a compliment if I say that the result of our meeting may be the most grateful service I can render to my country; and I doubt not that you, equally confident of success, will feel convinced that it is only by repeated triumphs in even combats that your little navy can now hope to console your country for the loss of that trade it can no longer protect. Favour me with a speedy reply. We are short of provisions and water, and cannot stay long here.

It's worth noting that Lawrence never actually received the challenge, but sailed out of Boston harbor anyway with the intention of fighting a straight-up duel with Shannon. Broke had commanded the Shannon continuously for seven years at the time of his engagement with Chesapeake, and had worked tirelessly with his crew on gunnery and cutlass drill. The Chesapeake had its wheel shot away and Lawrence was mortally wounded early in the engagement. Broke boarded with his men and Chesapeake surrendered after a very short fight, lasting only about 10-15 minutes. The Americans suffered 48 killed and 99 wounded out of a crew of 379, while the British crew had 23 killed and 56 wounded of a crew of 330. Broke himself was among the wounded, having received a cutlass slash to his head that left him unable to serve; he became a gunnery specialist and died a rear-admiral.