Around the world, across many centuries, traditional festivals feature the portrayal of traditional characters unique to the occasion. Many of these have ancient roots, though every one can be seen to have changed over time. Such characters are, in general, anonymous (or at least able to step out of their performers' everyday selves), tied to the context in which they are seen, and consistent from one instance of the event to the next (even if the performers change as years go by); most are associated with dance or other performance.
One example we've looked at in passing already is the hobby-horse who accompanies the Abbots Bromley Horn Dance in England. Hobby horses are a part of many traditional English folkways, and they're often presented in very different contexts. The two 'Obby 'Oss figures who appear at Padstow, Cornwall's May Day celebration, for example, represent a wild carnivalesque figure (the "Old 'Oss") and a more subdued approach to the ceremony (in the "Blue Ribbon 'Oss," sometimes known as the "Peace 'Oss" after WWI but first introduced by the Temperance movement in the 19th century as an alternative to the often drunken band that accompanies the Old 'Oss); each parades through town with a group of "teasers" in festive costume. It seems hobby horses used to be closely associated with morris dancing as well; Ben Jonson describes the sorry state of an early 17th-century morris side that doesn't even have one. To Jonson, a proper morris side had bells, a hobby horse, and a fool to accompany them. (Most modern morris sides omit the fool and the horse, though it's clear they were both still a normal part of the tradition at the time that Cecil Sharp collected their dances. It's likely that the revival has given them less emphasis precisely because Sharp saw the dancers as the important part and spent very little effort writing about the special characters.)
It's also common for rapper dance teams, which dance primarily for festive occasions in northern England, to be accompanied by a "captain" (also called a "tommy") in formal attire and often also by a "betty", an obviously-male dancer in drag. Most longsword teams, performing the more formal and older style of English hilt-and-point sword dance, also employ a captain, though many favor a more military look for how theirs is dressed. Interestingly, we can document the evolution of the captain figures over the course of the 19th century; they clearly originated as fools. (Many German sword dance sides now use a fool in a similar role.) The "betty" clearly began as a Maid Marian, another common character in England - in fact, it's believed that the Maid Marian of Robin Hood legend originates from both characters being a prominent part of springtime festivals in early modern England. In that context, the Maid Marian role was always played by a man in drag, usually bearded, until her redefinition as the love interest of Robin Hood. (Sharp describes some of the cross-dressed characters in dances he documents as still being identified as "Maid Marian," though the only one I know of that retains this appellation today is the one who accompanies the dancers at Abbots Bromley.)
In Germany, it's common for towns to have a wintertime festival, most often at shrovetide but sometimes at other times of the year, that features a strohbar or "straw bear" - a man covered head to toe in straw who dances through the streets. (This procession appears to originate in the 19th century, but may have earlier antecedents.) Like the Cornish 'Obby 'Oss, the straw bear must be accompanied by attendants, their primary function in this case being to act as guides - the bear dances and cavorts, but has no sense of where to walk, as in many instances the costume fully covers his face with straw. This also appears to have been traditional in a few English towns at one time, though only Whittlesea has revived the custom.
The straw bear is sometimes seen as an analogue of a uniquely modern English figure, the Jack-in-the-Green. This is a character uniformly associated with May Day, when in the early modern period people decorated themselves with garlands of foliage to mark the season; over time some participants got more elaborate about their attire, and eventually a custom began of certain trade guilds competing to dress the most elaborately green parade participant. By the late nineteenth century the modern form of this had developed: a wire or wicker framework, usually conical or pyramidal in shape, that completely covers the wearer with foliage from about the knees up, leaving only a hole for the face. The point is usually several feet above the person's head, and the exposed face is covered in green greasepaint. As another strange and unique example, South Queensbury (near Edinburgh) has a custom wherein the "Burryman," a local covered head to toe in sticky burrs, parades through the streets moving in the awkward way permitted by this costume. The origins of the Burryman are unknown, but the local market fair with which it is associated dates back to 1687 and it is said that the Burryman has been a part of it from the beginning.
Many Central European people apparently once had a custom, documented in Frazier's The Golden Bough, in which a grain spirit was said to live in the grain. It was often seen as associated with an animal, commonly but not always a wolf; in some villages, it was apparently the custom that whoever harvested the very last sheaf of grain would then immediately and temporarily shed their everyday identity to impersonate the wolf, who would be set on by pursuers with the notion that whoever caught them would have good luck. This custom lacks the elaborate, semi-planned and artfully-constructed nature of the other characters, but in this it becomes an accessible way to assign the traditional role in a random but symbolically resonant manner.
These characters, unique to their specific festivals, have an echo in the modern American custom of mascots; not only are they a part of most athletic events both here and abroad (with each one representing the home team and distinctly associated with the games hosted at their venue), but many civic festivals in America have one. For example, in my neck of the woods there is a park that holds an annual "frog fest", which is attended by a costumed "water quality frog." However, these characters are uniformly modern inventions, part of a wider tradition but in each case a newly-invented figure introduced from the top by event organizers. The origin of specific characters in more traditional contexts is less clear, though clearly somebody invented each one; the Jack-in-the-Green is perhaps the best understood development, however, and we know that to have been more or less unplanned by the people who originated the modern Jacks' precursor customs.
Another modern-day echo, of course, is Santa Claus. Unlike the standard folk festival characters of Europe, Santa (and other childhood legends such as the Easter Bunny or the Austro-German Krampus) is primarily a mythical character who happens to be represented in costumed form for some seasonal events. The hobby-horse, straw bear, and Jack-in-the-Green have no perceived existence outside the festivals at which they take center stage.
One can also draw a connection with rodeo clowns. As event staff, these serve a specific function - they play a safety role, protecting the audience from any out-of-control animals - but have also developed a highly specific style of costume and performance, again associated only with the occasion of the rodeo. (Much like English and German market fairs, which are held on different days in different towns so that many of the same traveling vendors can hawk their wares throughout a county or two, rodeos tend to happen once annually in each location, though for the participants they are a year-round occurrence.)
Other characters are not associated with any specific occasion, but have a similar origin. For example, it has been postulated that some of the masked characters of the Italian commedia dell'arte tradition originate as advertising characters by which mountebanks drew attention to themselves for purposes of selling their wears in late medieval and early modern Italy. Over time, these characters came to be of greater interest and potential profit than the salesmanship and hucksterism, and so they evolved into a purely theatrical performance style. The fools that accompany many folk dance sides are similar - they are part of whatever occasion happens to call for a dance, rather than being as seasonally-associated as the winter straw bear or the spring Jack-in-the-Green.
Intriguingly, the association with certain characters and seasonal observances has reasserted itself in recent centuries in the English tradition of the pantomime. While these plays don't have the same characters from year to year, they adjust the characters to fit certain standard types - and, in their early days in the 19th century, the main show was always followed by a "Harlequinade," which did involve stock characters associated only with this Christmas custom. Intriguingly, two of the prominent characters from the Harlequinade, Harlequin and Pantaloon, derive directly and obviously from Arlecchino and Pantalone, two of the oldest and most ubiquitous figures from commedia dell'arte.
Chinese lion dances and Cherokee booger dances involve performers who serve a similar role to their European counterparts. In each case, at specific festivals, select participants take on the role of a character and dance, in a traditional style, for the amusement of others in attendance. In a lion dance, a pair of highly-athletic dancers perform impressive feats while wearing a two-person costume depicted a highly-stylized Chinese "Lion"; while the dance may have its origins in seventeenth-century dynastic politics, today it is associated with Chinese cultural festivals and especially the New Year. In the Cherokee booger dance, a large group of masked performers dance with the other participants about a bonfire; the masked participants portray exaggerated lewdness, often directed toward all females in attendance. While the identities of the booger dancers are well-known (it's most of the men), often no individual performer can be identified, thanks to their masks. There are other examples of similar customs originating in other parts of the Americas, in other Asian festivals, and in Africa, as well.
The cultural contexts of these performances are quite different from the Germanic characters, obviously, or from such other Western examples as American mascots and Italian commedia. This disparity of cultural significance despite a few points of commonality highlights the fact that anonymous or semi-anonymised performers portraying idealized representations of an occasion seem to be a widespread part of many human cultures, regardless of the significance that may be attached to them in any given case or to the origin of any particular observance in which they participate. The characters evolve and the performance styles change, but the impulse to watch a distinctive and comic seasonal figure, it seems, is eternal.