Traditionally, these artistic sarees weren't sold, they were often women's dreams and diaries, although the art form can be bought easily nowadays. The birds represented freedom, parrots were associated with (secret) love, roosters (and hens) with domestic abundance, and peacocks with royalty. White was purity, terracotta strength, and green, nature; the tree of life. The flowering vines, the interconnectedness of life, the fence a border (protection or prison, depending on one's perspective), and running stitch, the constant reliable flow of a river's waters. The bird's eyes were often sewed on last, bringing the birds to life, just as idols are often animated.
Read a Nakshi Kantha correctly, and you'll probably learn more about the embroiderer than you would through hours of conversation with her.
And these diaries, they're safe — they're devoid of meaning to most.
Reading specifically into this kantha motif of paired birds drawn from a verse in the Upanishads which itself apparently echoes an earlier line in the Rig Veda:
द्वा सुपर्णा सयुजा सखाया समानं वृक्षं परिषस्वजाते ।
तयोरन्यः पिप्पलं स्वाद्वत्त्यनश्नन्नन्यो अभिचाकशीति ॥ १॥
(Hopefully, I've not made a mistake; there's little to recommend my familiarity with Sanskrit.)
The two embroidered birds, one barely visible, ensconced in the tree of life, are a reassuring retelling of the old verse; one tastes the fruit of life, the individual soul (Jivatman) and experiences the fruits of karma while the other, the Supreme Witness (Paramatman) looks on, a detached witness. One is bound to the fabric of life (and the cloth) while the other remains essentially free, a constrained material world against an unconstrained divine one.
That the birds are peacocks is suggested by their crests and elongated stylised bodies, and their being peacocks, believed to neutralise toxins, transmutes worldly suffering into spiritual wisdom. This is the ultimate liberation, the individual ceases to be a victim of circumstance and remains, in some way, incorruptible as it merges with the eternal, unchanging consciousness of the divine even while embedded in the material.