I just finished the drama yesterday and noticed there hasn't been a reply to your question. I was also intrigued by them and my conclusion of the bands is that they are very symbolic and special to Lee Yeon. It serves as a link that connects both him and Aeum (his past love). The bands represent how they bind him to his love as well as a sad reminder of the debt he had to pay upon Aeum's request. It is an object that serves as a memory as well as an attachment of his love for her.
After they met, and he realized that Ji Ah is Aeum's reincarnation, he stopped wearing the bands since they started wearing couple rings. Just like in the drama, he said that Ji Ah isn't a shadow of his past so he decided to look at her properly in the present timeline. But you could see in Season 2 of 1938 he's back to wearing the two bands again. It's basically sort of like a "red thread" of destiny for him, reminding him of his enduring and unyielding love for Ji Ah (Aeum). Something he can look at whenever he misses her you could say.
Hope that helps even though it's a late reply.
