Purple in the ancient Mediterranean means one specific dye, and the reason it signified power is that making it was appalling.
The dye comes from marine snails, principally Hexaplex trunculus and Bolinus brandaris, and from a single small gland in each animal. The secretion is not purple when taken; it is a pale fluid that develops colour on exposure to light and air. Producing a usable quantity required enormous numbers of snails, and the processing involved crushing and prolonged salting and simmering. The stench of the dyeworks is remarked on by ancient authors, and the mounds of crushed shells at Tyre and Sidon are still identifiable archaeologically.
Pliny describes the process and its cost. Roman sumptuary law restricted the deepest purple to the emperor, and Byzantine practice went further, with the phrase born in the purple referring to a chamber of the imperial palace. What the law was regulating was less a colour than a demonstration of expenditure.
The industry did not survive the fall of Constantinople in 1453, and the technique was genuinely lost for centuries. Modern reconstructions have established the chemistry, and the principal colourant is dibromoindigo, chemically a close relative of indigo.
The point that matters for reading the sources is that Tyrian purple was not a fixed hue. Ancient descriptions place it anywhere from a deep red to a blackish violet depending on species, concentration and process, and the modern mental image of a clear bright purple is probably wrong for most of what the texts describe.