Woodman in his "The History of the Ship" states that by the time of the Napoleonic Wars, naval warship designs across most of the European countries were very similar, citing several cases of the British navy submitting orders for exact copies of captured French warships which had exceptional characteristics. A specific example is the French ship "Pomone" which was captured by the British Navy and promptly copied with the "HMS Endymion" in 1794. Such copying of the enemies designs was a deemed a necessity in order to maintain technological equality during the war. "James' Naval History" which is a collection of contemporary accounts of British naval battles from the Napoleonic period, repeatedly notes that strange sails were sighted and only upon bearing down upon the unknown ship were they able to determine its status as friend or foe. This would imply that the similarities between the sail designs were, for practical purposes, indiscernible when "hull down" i.e. only the sails are visible above the horizon.

Even at closer ranges identification could range from difficult to impossible. While there is record of experienced sailors being able to notice differences in hull design between French and British warships these differences are not usually enough for proof positive identification. The British navy enacted a system of flag signals which required specific signal responses, in order to aid in identification. These signals rotated regularly to ensure that secrecy was maintained incase the signal books fell into the wrong hands, along with strict guidelines to ensure that the lead covered signal book was discarded if the ship was in danger of being captured. While accepted conventions of warfare prohibited the firing upon an enemy with out flying the national colors it was not unheard of for fake colors to be presented for the purposes of fooling the enemy for a time.

As another example of the apparent similarity of British, French, and Spanish ship design, "James' Naval History" reports an incident (I lost my place in the book so I don't have the ship names presently, will update), in which the British Naval ship bears down on an unknown ship which when hull up still refuses to answer signals or throw out colors, this prompts the captain to close within gun range and fire a shot over the bow of the fleeing ship, thus beginning the battle. It was noted even when within gun range the British could not determine the nationality until the enemy ship flew her French colors at the commencement of the battle.

In closing, while there is record of ships being identified by unique rig and design in the early 1700's by the Napoleonic Wars the science (or perhaps more precisely the art) of ship making had refined to the point of convergence in ship design and rigging, a claim further supported by the apparent difficulty in identifying uncooperative ships.

I hope this answers your question with enough specificity, if you have any questions feel free to ask and I will help with what I can.

Edit: I took the question to mean literally "Ship rigged" naval vessels. When examining smaller vessels used for scouting, patrol, and courier work there were absolutely different rigs and vessel designs because many of these smaller coastal oriented vessels were based on the regional fishing and trading vessels of the time. Thus British Cutters tended to wear a single mast with a large gaff rigged sail extended to the rear of a fore and aft rig with topsails, while many Spanish vessels in the Mediterranean used a Xebec rig with multiple masts and large "Lateen Rigged" sails.

So, there is a simple answer and a complicated one to this question. The simple answer is that there really wasn't a lot of change and innovation in the hull design and rigging for a LONG time (from the mid-1600s through the Napoleonic war), but individual navies had distinctive characteristics largely derived from cultural preference and natural resource. So just as today we might draw conclusions on whether a car is German or Japanese or American based simply on "style," the same ideas held true amongst the ships of the time.

Also, unlike modern wars where destroying enemy ships was common, many vessels of this time lasted for decades, sometimes being captured, recaptured, and recaptured again. The HMS Victory was ordered in 1758, launched in 1765 and was in sailing service until 1822. Adding to this, the (British) Royal Navy was deeply conservative, suspicious of innovation and rooted in tradition, so it (more so than other nations) was slow to change, making it easier to spot differences between the Royal Navy and other countries' war ships.

By contrast, during the Napoleonic Era (which again, is a pretty long time) most other protagonists adapted their entire naval strategy several times depending on the objective, plus the alliances with other countries (Spain, for example) shifted from friend to foe several times. So you had a long, constant, conservative and (generally) single-purpose player in the British, confronting upstart/changing/more-technologically-aggressive opponents in the French, Spanish, and Dutch.

The more complicated answer is... physics! The entire purpose of a sail plan and rigging is to move a specific shaped hull through the water. And this was often quite unique to the individual vessel rather than the propensity of the captain. You don't just have to contend with the strain on the sails from moving forward but also the fact that most of the time the wind is from the side, the weather may be shifty (depending on station), cargo, water ballast, height of the guns above the waterline... oh, about a zillion things. While the British, as essentially an "industrial" navy, focused on maintaining continuity and compatibility with a fleet spanning the globe, pushed toward standardization (even when it meant eschewing innovations), other navies had more flexible priorities. Example: the most famous US warship of the period, USS Constitution, was an innovation "between classes" of the British system.

Take the battle between the USS Constitution and HMS Guerriere in 1812. Both were "frigates," but the Guerriere carried sixteen 32-pound guns and thirty 18-pound guns vs. the Constitution's thirty 24-pound guns on deck and twenty-four 32-pound guns in the ship proper. Simple math illustrates the difference: Guerriere could throw 526 pounds of metal with each broadside, Constitution could throw 744. That's projectile weight so imagine how much heavier Constitution would have to be to carry guns capable of throwing that much weight.

Long story short, every nation involved even tangentially in the Napoleonic wars had different priorities (think "protect my harbors" versus "rule the high seas" versus "get our landing craft over to England") and therefore designed different kinds of vessels toward their goals. Simple physics dictates that those different types of vessels have a different sail plan, which would be as easy to spot by a trained observer as the difference between a WWII bomber and fighter (where there are published charts on how to tell the difference).

And if you really want to nerd out on this (and since you're on this subreddit, of COURSE you do) there's a lovely research paper on the differences in rigging between French, Dutch, and English ships of the (early) era. Pages 153-158 are of particular interest.

And as a postscript, it's also important to consider spy networks and destinations. If a French/Quebecois spy in Halifax, Canada noticed an English ship leaving for Caribbean duty, word would find its way to those areas to be on the lookout for that ship. And not just the sail plan, but the condition of the sails would be a giveaway as to the identity of that particular vessel once it arrived on station.