The Zhuangzi was never scripture in the sense of a rule of life, and Confucian scholars were mostly wary of it. Its authority came from being read, endlessly, by everyone educated. In 742 the Tang emperor Xuanzong gave it the honorific title True Scripture of Southern Florescence and its author the title Perfected Man of Nanhua, formally installing it in the Taoist canon, but its influence had never depended on that.
Its first great period was the third and fourth centuries CE, when the Xuanxue or Dark Learning movement made it, with the Tao Te Ching and the I Ching, one of the three texts through which educated men discussed reality. That coincided with the arrival of Buddhism, and the Zhuangzi supplied much of the Chinese vocabulary in which Buddhist ideas were first rendered and misrendered. When Chan Buddhism took shape, its characteristic moves were already familiar from this book: distrust of discursive argument, teaching by paradox and anecdote, the master who answers a metaphysical question with something absurd, skill acquired until it becomes spontaneous. Chan's debt is to Indian Buddhist doctrine and Chinese literature together, and the literary half is largely Zhuangzi.
In poetry and painting the effect is at least as large. Tao Yuanming's retirement to his fields, Li Bai's flights, Su Shi's essays on the relativity of large and small all draw on it. The landscape painters took from it the conviction that technique should disappear into the act, Cook Ding with a brush. The recurring figure of the recluse who declines office, which shaped a thousand years of Chinese self-presentation, has its sharpest source here in the story of Zhuangzi refusing a ministry, preferring to drag his tail in the mud like a living turtle.