Paisley as we know it today is based on a floral motif called a boteh. Originally, this was fairly clear - as in this example of a strip from an early eighteenth century shawl, the boteh was fat and clearly made up of flowers, with a slight twitch to one side at the top; if you zoom in a bit, you can see a patch of ground beneath it and stems rising up and out. The boteh became more and more stylized, with a narrower width and more pronounced lean and swirl at the top, as you can see in this late eighteenth century example, this whiteworked shawl, and this yellow shawl - though I would hasten to note that there are also counterexamples of plain, straight boteh, like this mid-nineteenth century shawl. But by the 1840s, it was fairly common to see paisley as we know it today - a decontextualized swirly shape.

Why did this happen? The whole thing is kind of funny.

The original Kashmir shawls we're talking about that were brocaded with paisley boteh were being made by Indian manufacturers, to be purchased by both locals and European import/export merchants. Western tastes soon took a turn for the shawlier - with the fashion for Neoclassical-inspired dress, shawls became hugely important accessories. (Greek and Roman painted vases, statues, and reliefs very often showed women with large enveloping or draped shawls.) As the market became bigger, it was worth the manufacturers' time to change up the patterns for the Western taste, which found larger and attenuated boteh more exotic and aesthetically pleasing. (Here's a shawl that uses a boteh for a local audience.) Western manufacturers also began to imitate the style in various mill towns ... like Paisley in Scotland, from which the pattern gets its English name. (And the importation of Western examples by Indian manufacturers made it possible for more copying to go back and forth.) The motif became more and more stylized until it had nothing to do with the original.

A while back, I was told that an old Persian interpretation of the textile pattern was that it represented “the universe moving around you.” Essentially, the point of what I heard was that it celebrated individualism and self-identity.

I can't find a reference to this anywhere, and it seems very unlikely. Individualism and self-identity are not highly celebrated ideals, historically - they're very tied to Western society in and after the twentieth century.