Is it context and history or just that real game positions have much lower entropy? Most people can easily remember a sentence after a quick look, a similar-length string of random characters a lot less so.

I think this is correct. I play Go at a strong amateur level, and on the 19x19 board, I can often fully recall 200+ move games against "normal" opponents, because the moves tell a story that makes sense. But if I play someone much weaker or much stronger who makes a lot of unexpected moves, I'll have trouble remembering those moments.