You will not be surprised that the words wage and wager are related. They look and sound similar, even if they mean different things. More surprising is the fact that the word wed is related to the other two words.
It seems that the concept the three words have in common is not money, as you might assume from the first two words, but a pledge. The Old English word weddian comes from the Germanic base of the Scots wed, meaning "a pledge." When couples marry, after all, they make a pledge to be true to one another.
The words wage and wager came from the same source. An employer pledges to pay a wage in return for labor performed. When a gambler makes a wager, he pledges to pay his debt should he lose.