Most likely because of their origins.

In the eighteenth century, western women typically wore neck handkerchiefs to cover their chests over the typically low-cut gowns of the day. (Here's a good image of a woman in one: Portrait of the Brak Family, 1752.) Having a linen or muslin handkerchief of a snowy white was preferable from a class standpoint: having clean whites on your person, whether as a neck handkerchief, stock or cravat, or the ruffles from your shirt or shift peeping out under the sleeves of your outer garment, showed that you were a clean and tidy person. (I discuss white underclothing in this previous answer, if you're interested.) The white was also an excellent setting for delicate whitework embroidery, you can see on this lovely example at the Metropolitan Museum of Art.

However, women of the lower working classes (doing manual labor, selling wares in the market, etc.) did not have delicate whitework and often did not care about the ideal purity of white linens. "The Benevolent Cottager" shows one such woman, dressed in stays and petticoat without a gown over them (despite the stereotype that of course working women would do without stays, they would in fact typically take off the outer upper garment first instead). Around her shoulders, she has a green and yellow checked neck handkerchief. A patterned or printed one would be less likely to show sweat and dirt immediately, and the fabric might also be cheaper to buy. Though I don't want to give the impression that only poorer women wore colored handkerchiefs! An affluent or wealthy woman who was traveling, privately spending time at home, shopping, etc. might also have preferred one that wasn't snowy white, for practicality or to signal that she was dressing down.

At the end of the seventeenth century, the west went mad for calico: printed cotton from India. It was not the bottom end of the textile market, but it was no longer expensive by this point, and therefore practically everyone could get in on it. The French and English governments passed laws against using imported calico in order to protect the domestic cotton-printing industry; many people ignored them. Calico was also seen as setting the social order into disarray, because so many different people could be found wearing the fabric. It was sold to be used as coats, gowns, petticoats, upholstery, bed hangings, mantles ... and handkerchiefs. Printed silk handkerchiefs were also popular relatively-cheap imports, and would continue to be throughout the century, like calico. Here is a printed calico handkerchief from Pulicat; here is a silk one from Murshidabad. The word "bandanna" was in use for these handkerchiefs, although it technically referred to ones that had been tie-dyed, with little white circles on the body of the design.

Okay, now where is the connection to paisley? Well, the Scottish town of Paisley was a center of imitation Indian shawls, as I discussed in this previous answer: What was the original meaning behind the Paisley pattern?. The stylized boteh from imported Indian shawls became the motif we today call "paisley", and at the time was an important part of signifying that something was Indian. While we now no longer have this association, it was for some time a deliberate one: printed neck handkerchiefs are meant to be from India, therefore they should have paisleys on them.

(One source I referred to is open access, and may be interesting reading if you liked this: "The American Market For Indian Textiles, 1785-1820: In The Twilight Of Traditional Cloth Manufacture" by Susan S. Bean of the Peabody (now Peabody-Essex) Museum, from the 1990 Textile Society of America Symposium.)

Fantastic response to a random question! Thank you!