[red_admiral]

The rule of three was actually good and common writing advice in the pre-LLM era. There's psychological studies that show 3 is a good number to grab human attention, which is why you have "an Englishman, an Irishman, and a Scotsman walk into a bar" jokes. Even my Algebra professor said that a well-written definition should have three conditions, for example a group is a set with an operation that is (1) associative, (2) has a neutral, and (3) has inverses. I don't think he was completely joking.

(For stories with multiple protagonists, the common choices that seem to work best for readers are 3 or 5. Humans are weird.)

I suspect that LLMs use that rule so much, because it's so common in their training data, for good reasons.

[JuniperMesos]

A group is an algebraic structure, and many algebraic structures are defined as sets with an operation that obeys some number of axioms. It happens that a group is a structure whose operation has exactly three axioms; but a monoid for instance is a set with an operation with just two axioms - it's a group without the "has inverses" axiom - and I don't think this makes the monoid worse than a group in any meaningful way.