"Checkmate" comes from Persian

>شاه مات‎ (šâh mât, “the king [is] amazed”). Perhaps conflated with Arabic مَاتَ‎ (māta, “to die”).

The origins of the game itself are a bit blurrier, but not four thousand years old blurry. But, there were games four thousand years old that if we went back in time and saw people playing we would probably describe them as "like chess" as they had similar elements.

There's a lot of games from antiquity that we would recognise or be somewhat familiar with, and probably pick up the rules of relatively quickly. https://www.joshobrouwers.com/articles/ancient-greek-heroes-play/