This is a far-fetched theory I have | will change ofc as the drama progresses

Let's call Hideo's adopted mother "Mrs Sato".

So we know Mrs Sato was from Joseon, and has been killed by Sato Senior. But he told a lie to Hideo and Hanako, that their mother ran away. But in actual, Sato Senior discovered she was part of the Joseon freedom movement and had fooled him all along. He naturally kills her but records the event as her running away back to Joseon, probably with another man - making him a figure to sympathize with raising 2 kids as a single father while doing his duties for Japan.

Flashback:

Before becoming Mrs. Sato, she was the biological mother of Ga Kyeong. She was part of the same freedom movement that So-ran's husband was part of. Either her husband was killed or not in the scene, but Mrs Sato was also a spy. She pretended to be in love with Sato Senior (who went by Ishikawa at that time).

Then the events of 13 year ago happen - and So-ran's daughter Keum So-hee is killed. Maybe Mrs Sato finds the pendant after So-ran escapes OR Mrs Sato helped in her escape and somehow came across the pendant. In any case, its in her possession now. Given she was also part of the freedom movement, she keeps it for sentimental value or for motivation that she has to continue being a spy and pretend to love the man who massacred her entire village.

But then Sato Senior is called by court, and he must leave for Japan. In parallel he has also found Hideo [the incident we saw of him pickpocketing] or before this massacre. Basically, he has Hideo with him, and then asks Mrs Sato also to leave with him. Therefore, Mrs Sato leaves with him leaving behind Ga Kyeong with her relatives. But before leaving her - she gives one-half of the pendant to Ga Kyeong and also goes to the photo studio to take the picture together [which we saw in flashback in Ep 5]. She takes the photo with her [or maybe it's this photo that she has kept at the Jongmo pawn shop - hoping to come back and take it - along with some other things so must be some kind of storage box to keep it safe, though am not sure of this]

Now she has one half of the pendant with her and lives in Japan with Sato Senior and Hideo. Soon she gives birth to Hanako. Now she probably was sending information back, and Sato Senior starts getting onto her. So she gives the pendant to Hideo as a keepsake, knowing her end is near, or feeling she must flee.

Hence this is how they both have the locket - each half. At least that's what I am thinking....

Present

This way they will together discover the pawn shop, get the photograph and Hideo will put the pieces together. He already knows that he's the adopted son - and so it would not be weird for him and Ga kyeong to be together - the motherly act would be of Mrs Sato's pendant being found by them and in a way So-ran bringing them together, first through deception but ultimately by fate.