I suspect (in a probably ignorant fashion) that this is because learning process has been reading a lot of algebraic chess notation (such as "1. e4 e5 2. Nf3 f6 3. Nxf6 gxf6 4. Qh5! +-") then, to play, generating more of it without considering the rules of the game. This is exactly how it's always felt to me when playing chess against LLMs. Sure, "1. e4 e5 2. Nf3 Nc3" looks innocent to somebody simply learning the syntax of algebraic notation, but that Nc3 by black is an illegal move.

An LLM is the wrong approach for playing chess.