When Anju gives Link her letter to deliver to Kafei, she's worried that Kafei has skipped town to back out of their engagement. (She might even wonder whether he's still alive!) Kafei does want to go through with the wedding, but he can't show himself because the Skull Kid transformed him into a child and Sakon stole his Sun Mask. It's significant that Kafei gives the pendant in response to the letter—it means that Anju will understand it as if he'd written an actual note in return.
For Anju, the Pendant of Memories says "I'm safe, I haven't abandoned you, and we're still gonna get married." It's a little bit similar to an engagement ring or a promise ring in this way.
But I don't think the Pendant of Memories is supposed to have a cultural significance in the realm of Termina, or refer to a tradition in real-world culture. We see in Anju and Kafei's storyline that in Termina, married couples symbolize their bond by crafting masks—Kafei can't go back to Anju until he's recovered the Sun Mask he made that was stolen by Sakon. We can imagine a different game where Kafei can't see Anju for some other reason (i.e. still insecure about being four feet tall) and gives Link the Sun Mask to deliver: The message, for Anju, would be the same.
Without the culturally appropriate symbol of love with which to send this message, Kafei gives Link the Pendant of Memories—which in the Japanese version was called Kāfei no Tarekazari, Kafei's Pendant. He's been wearing the pendant all the time up to this point; it's a distinctive piece of jewelry that Link couldn't have gotten from anyone but Kafei.
It's a little bit like Kafei giving Anju his dog tag or letterman's jacket, but the token is far more unique than the latter and far more romantic than the former. When she receives it, she knows Kafei still loves her, and she knows he's serious.