I'm not a scholar of piracy specifically, but I've studied Jamaica and New York pretty extensively in the late seventeenth and early eighteenth centuries, and piracy (and its cousin privateering) come up a lot in that period, as you might imagine. And the short answer to your question, from that documentary base, is that a one-on-one pirate-merchant combat is reasonably realistic.

One caveat: since a lot of the tropes and historical inspirations for pirate movies actually come from privateers, there's some clarification needed. Privateers are granted a license by their government to employ a ship or ships against merchant traffic (and other targets) of their government's declared enemies. They're much more common than pirates, who aren't licensed by anybody and are theoretically "enemies to all." However, as you might imagine the distinction could be easily blurred, depending on how official the war was (Henry Morgan's problem in the 1670s), how stringent the captains were about pursuing only enemy targets (Kidd's problem in the 1690s), and how reluctant the crews were to come in after the war ended (everyone's problem). I'm going to answer the question with reference to both, and try to make distinctions where they seem applicable.

In the context of the English/British colonies, merchant convoying was pretty uncommon outside of periods of declared warfare between the major maritime European powers. However, that does cover the years 1689-1697 and 1701-1713 at least. But plenty of ships didn't convoy even during war, either out of a desire to avoid slower travel or out of a lack of warships available to do it. And we know ships traveling alone were periodically captured by pirates or privateers--for instance, Robert Hunter's ship was captured when he was sailing to Virginia to take over as its governor in 1707.

In peacetime, the most well-known convoys are the Spanish treasure fleet and the Mughal hajj fleet, and there weren't many regular ones besides that, although convoying increased in the early eighteenth century generally. Pirates and privateers often took advantage of the lack of protection. That's what "the Red-Sea men," as the governor of New York called them in 1698, were doing there and throughout the western Indian Ocean--attacking unconvoyed ships with rich cargoes (although the biggest success was actually an attack on a convoy). Incidentally, a number of those guys were proper pirates, operating without any commission and attacking commerce relatively indiscriminately--but some, like Kidd, started out as commissioned privateers and only (arguably) slid into piracy afterwards. And again in 1718, we see Robert Hunter, now a little wary of being captured, telling the board of trade that he intends to wait for a ship of war to sail home from New York, "the pirates being busy on our coasts" where there were no convoys, or the duke of Portland complaining in 1723 that the two last ships he'd sent with letters from Jamaica had been taken by pirates.

But what about pirates and privateers themselves? Did they travel in groups? Well, sometimes yes and sometimes no. From the documentary evidence with which I'm familiar, privateers issuing out of New York and Jamaica did sometimes travel, at least initially, in groups. For Jamaican privateers especially, taking a crack at the Spanish treasure fleet or any of the big targets along "the Spanish main" required a group. Morgan's successful 1668 raid on Porto Bello, for instance, featured at least twelve ships. But privateers frequently traveled solo as well, at least partly because that was the best way to ensure that one captain and crew retained all the plunder and profit from a capture. As with privateers, pirates' most famous work was in groups, out of "bases" like Madagascar or New Providence. Henry Avery captured a massive treasure haul from a Mughal convoy with a small pirate flotilla in 1695, John Taylor and Olivier Levasseur took an even larger prize in 1721, and the ur-pirate (for pop culture), Edward Teach, seems almost always to have sailed in a group. But plenty of pirates did sail on their own out of those bases, like Shelly out of Madagascar or Calico Jack Rackham out of New Providence, and some toggled back and forth between the two, like Bart Roberts. And there were, of course, pirates who didn't come into contact with the bases and did their own work solo.

So, after all that, though the most well-known privateering and piratical exploits involved groups of vessels on both sides, it is very realistic for a movie to depict a lone pirate ship capturing a lone merchant vessel.

Sources (besides the primary documents):

Marcus Rediker, Villains of All Nations

Mark G. Hanna, Pirate Nests and the Rise of the British Empire

Hilary McD. Beckles, “The ‘Hub of Empire’: The Caribbean and Britain in the Seventeenth Century,” in The Oxford History of the British Empire: Volume I .

Richard S. Dunn, Sugar and Slaves: The Rise of the Planter Class in the English West Indies, 1624-1713.

Robert C. Ritchie, Captain Kidd and the War Against the Pirates

Not my area of expertise as per history (which is focused on central Europe in the 1920s, 30s), but I sail a lot and I think the technical considerations of sailing need to be mentioned.

Visibility on the water is about 3 miles on the best of days. That is with good light, no fog or haze, no rain, etc. Most of the time it's less. On top of that, there is the question of the weather. You need to be relatively upwind of your quarry to catch them. It's not a trivial matter to 1) find another ship on the open water 2) be in a position to catch it. Participating in sailing races today shows you just how important it would have been to be in the right place and right time.

Having a convoy of ships in the age of sail only helps a bit. Different ships sail differently and it would be more than possible for a quick pirate sloop, which can sail toward the wind much more effectively than a heavy merchant, to sweep in, nab or sink on ship in a convoy, and sail off before anyone else can help.

For ex. a two merchantmen are sailing along at a brisk (for merchantmen) 4 knots with a following wind, 1 mile apart. A pirate has the weather gauge, sweeps in from a mile away at 7 or 8 knots, and catches the last ship within 10 minutes of making it clear he is chasing (by his actions of course, not signals). It takes the other ship 10 minutes to gybe around. Then, because they are a mile apart, it takes at least two hours to come up to help that is if she doesn't miss any stays - it is difficult to make more than 1 knot speed over ground against the wind. By then, the pirate has captured one boat, and is more than happy to take on the second.

The pirate is also maneuverable enough that he can come around multiple times and keep out of the way of any effective broadside. On top of that, a merchantman is carrying cargo, thus limiting the amount of guns she can carry. Gun crews also need lots of training to be effective.

The differences between sailing qualities of different rigs and ships are really striking.

Convoys make sense when they include fighting ships or heavily armed merchant ships (Indiamen are an example I believe, but I'm not an expert).

Patrick O'Brian has a great description of this in the first book of his series - of a xebec coming in and capturing a merchantman before the escorting navy ship can come up alongside.

From the pirate's perspective, it might be easier to sweep for ships in a convoy - you would dramatically increase your sweeping area - but this would automatically make it nearly impossible to coordinate an attack. For ex. with say 4 miles between, the wind at a nice 15 knots, you would sail at about 7 knots. Both pirate ships would sail toward the quarry, matching speed. They would close the distance only when one of them has caught up. This means the second ship would arrive at best a half hour after the first, and that is only if the first and the quarry have stopped. If the one waited for the other, they would risk losing weather gauge, and letting their quarry get away.

Such wide sweeping would also make it much easier to be spotted and either found, or for ships to let each other know which area you are sweeping and avoid it.

Sweeping strategies were employed, but by respective navies to find pirates, enforce blockades, find enemy ships, etc.

As far as I know, a common strategy for pirates would be, as often as not, to stay within areas of high shipping, masquerading as fishing sloops or small traders, and rush in to seize vulnerable, slow ships. This isn't really possible with convoys.

Again, I'm no expert on the history of piracy, but when reading about the history of boat types there are many references of the type "the Bermuda-rigged cutter was a particular favourite of East coast and Caribbean pirates for its upwind performance, speed, and versatility."

In contrast to the frigates and ships of the line featured in pirate movies, these are one-masted rigs, similar to most sailing yachts today (the Bermuda rig is prevalent today as opposed to the, then common, gaff-rig), and would be crammed full of fighters, ready to sweep in on unsuspecting prey and disappear just as fast.

If there is any naval historian out there who would like to correct me on any points I would be glad of it.