The story of David and Goliath is based on the idea of two champions fighting a battle on behalf of their armies, but there are many other examples of this in literature and film. My Pakistani friend once told me that the border guard strutting contest during the daily closing of the Indian-Pakistani border was based on that tradition as well.
Did this practice ever exist anywhere or at any time in history? Or is this a construction to exaggerate the deeds of legendary warriors?
Yes, but also no, leaning more to no, because literary culture is Like That. More can always be said, and further input is welcome from other areas of the world, so if anyone would like to put their oar in, please don't let this post stop you!
For the meantime, OP, I commend to your attention u/Iphikrates' looking at how the Greeks did this sort of thing, with a bit of extra analysis in this thread.