The chakras are genuinely Indian and genuinely old. The version taught almost everywhere in the world today is neither wholly one nor the other, and the difference is well documented.
The tantric sources, Hindu and Buddhist, from roughly the seventh century onward, describe chakra systems in detail. They also describe different systems: four centres, five, six, seven, nine, twelve, twenty-one, depending on the text and the lineage. The number was never fixed, because the chakras in those sources are not anatomical discoveries but structures generated in practice, visualised, constructed and used within a specific initiatory system, and different systems needed different structures. Buddhist tantra commonly works with four or five. The scheme of six centres plus the crown, which underlies the modern seven, comes principally from a sixteenth-century text, the Sat-Chakra-Nirupana.
That text is the one that reached the West. Sir John Woodroffe, writing as Arthur Avalon, translated it in The Serpent Power in 1919, and that book is the direct ancestor of essentially all Western chakra literature.
The colours are the giveaway. The classical sources assign colours, but not the rainbow: they give the centres colours tied to their elements and deities, and those do not run red to violet in spectral order. The rainbow correspondence, red at the base ascending through the visible spectrum to violet at the crown, is a twentieth-century Western addition, popularised in the 1970s, and it is now so universal that it is generally assumed to be ancient. It is not.
The same applies to the pairing of chakras with endocrine glands and with a psychological developmental sequence: both are modern syntheses, drawing on theosophy and later on humanistic psychology, and neither is present in the Sanskrit sources.