I feel like it's more of a trope than anything useful.
This is why tabletop games were made, as a bunch of Prussian Generals got bored due to how limited chess was, and made their own game called "Kriegspiel."
It doesn't teach you about military strategy in the strict sense of the term, but more in a general way.
I would say that chess teaches you to think ahead of your moves and possible countermoves of your enemy, and also it teaches you to be ready to sacrifice your troops for a win.
I'm not a soldier and neither a proficient chess player though, so those are just my assumptions.
It's basically a narrative shortcut to show a character thinks "tactically". And also gives said character a chance to use cliche dialogue talking about "pawns" and what not.
You should see the chess match from No Game No Life where the pieces have wills of their own and refuse to be sacrificed because virtually no one would obey an order that requires them to die, something chess doesn't take into account.
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