It’s the sort of grammatical rule that’s easy to remember: use between with two conjoined noun phrases, but among with three or more. It also has nothing to do with English grammar or usage, but that does not stop the learnèd ignorant from foisting it upon you. It is an example of the etymological fallacy. Merriam-Webster’s Dictionary of English Usage (link) quotes J A H Murray (the first editor of the OED):
[Between] is still the only word available to express the relation of a thing to many surrounding things severally and individually, among expressing a relation to them collectively and vaguely.
The editors go on to say
The OED shows citations for between used of more than two from 971 to 1885. 971 is the date the Bickling Homilies were composed (link). I took a look at the index. The entries for betweonum show that it is used four times as a postposition (probably more of a verbal particle), and a couple of times split with its complement coming between the two parts. For example:
þa cwædon þa apostolas to þæm folce, ‘Heo bið swiþor gestrangod be us tweonum þurh Drihtnes gehát’. p.143.ll.11f.
then said the Apostles to the people, ‘She shall be much more strengthened among us by God’s promise’.
In other words, pretty much since English has been written down, between has been used with more than three items.