No ancient source calls this pattern the Flower of Life. The term is a twentieth-century coinage that entered general circulation through Drunvalo Melchizedek, who taught his Flower of Life workshops from the mid-1980s and published The Ancient Secret of the Flower of Life in 1999. The framework that comes with it (the sequence from circle and vesica to Seed, Flower, Fruit and Metatron's Cube, the derivation of the Platonic solids, the star tetrahedron called the Merkaba), is his and his successors', not an inheritance from Egypt, Israel or Greece.
This distinction matters more here than anywhere else in the subject, because the two halves are habitually welded together: the pattern is ancient, therefore the interpretation is ancient. The first half is true and the second does not follow from it. An Assyrian mason, a Roman mosaicist and a Cairene tile-cutter all drew this grid, and there is no evidence that any of them shared a doctrine about it, or held any doctrine about it at all beyond the facts that a compass produces it easily and it is pleasant to look at.
The strong modern claims (that the figure encodes the fundamental patterns of creation, that it contains the information of every living thing, that it is a record left by an earlier civilisation), are metaphysical statements. They are held sincerely by a great many people and they are the reason the pattern is popular now. They are also about forty years old, they are asserted rather than demonstrated, and they have no support outside the literature that asserts them.
Presented as what they are, they are a modern esoteric tradition with a real following and a real aesthetic. Presented as recovered ancient knowledge, they are false.