Moonstone's associations are almost entirely modern, and the exceptions are worth isolating from the general claim of ancient reverence.
There is a genuine Indian tradition. Moonstone is chandrakanta in Sanskrit sources and is associated with the Moon; in jyotisha it functions as a substitute or secondary stone for Chandra, the Moon, where pearl is the primary navaratna gem. It is regarded in Indian practice as auspicious and is a traditional wedding gift. That is a real, continuous and documented tradition.
The Roman claim frequently printed, that Romans believed moonstone was formed of solidified moonlight, traces to Pliny's discussion of a stone he calls selenites, which he describes as containing an image of the moon that waxes and wanes with it. Whether selenites is moonstone, or selenite gypsum, or something else, is not determinable, and the confident attribution to moonstone is a modern gloss on an ambiguous passage. It is the same name problem that runs through all ancient gem identification.
The rest (intuition, emotional soothing, new beginnings, travel protection, the sacral and crown centres), belongs to the twentieth and twenty-first century crystal-healing literature, which reached its present form through Art Nouveau jewellery's revival of the stone around 1900 and then through the metaphysical publishing of the 1970s onward. The chakra assignments in particular cannot be older than the Western synthesis of chakra material, which is late nineteenth and twentieth century.
Moonstone was not on the 1912 list. It appears as a June alternative in later revisions alongside pearl and alexandrite.