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

What we Know/can deduce:
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.

Theory:

Before becoming Mrs. Sato, she was the biological mother of Ga Kyeong.

(we know that Mrs Sato is going to be played by - Song Ji Hyo, and given this flashback in the Ep 5 photo studio - she seems like Song Ji Hyo, the lady who we can assume is Ga Kyeong's mother)

Assumption:

Mrs. Sato was a widow, we know this from Ga Kyeong's flashback when she's washing the clothes with other women. Assuming, she was part of the same freedom movement that So-ran's husband was part of. Either her husband was killed by Japanese or died of some other causes, but she joined the freedom movement in any case and her mission was also of a spy by becoming the lover of a Japanese - which was Sato Shinichi (who went by Ishikawa at that time).

Assumption 2:

Before the Kyeam-ri massacre, Mrs Sato (widow) and Ga Kyeong (child) used to also live in Kyeam-ri, Suwon. While on the outside Mrs Sato pretended to be unaware but she was in the same movement as So-ran's husband. She was not there when the massacre happened.

But she returns in secret, to have a word with the remaining freedom fighters, and learns about death of Soo-Hee and either helps So-Ran escape and So-Ran gives her the (complete) pendant as a thank you, or So-Ran finds it where Soo-hee is buried and keeps it with her as a reminder of the sacrifice of her people + motivation to continue fighting.

Assumption 3/Theory:
Post the events of the Kyeam-ri massacre, Japanese government had to intervene and Sato/Ishikawa and Fukada were called to Japan to be investigated. Since he had to go back to Japan, this is when Mrs Sato leaves behind Ga Kyeong with her relatives, earning the bad reputation that she ran away with a Japanese man, and maybe this is why they were bad to child Ga Kyeong making her an outcast. But before leaving, she gives one-half of the pendant to Ga Kyeong and wears the other half, as a symbol or parting gift.

She then also takes the photo studio, and gets the picture taken of them both - as a keepsake as she knows she has to leave her daughter behind and go to Japan. This photo she cannot take with her, and so she "deposits" it in the pawn shop along with other items possibly. The pawn ticket it what is found later, by Hideo and now in possession of the grown-up/present Ga Kyeong.

Assumption 4:

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.

Present Day - Theory: What can happen:

Given Ga Kyeong insisted to take Hideo and look for the Jongmo Pawn Shop, now that their feelings will grow (truly + part of their own plans) - they will find the shop and therefore also discover the photograph that Ga Kyeong took with her mother.

Now here - Ga Kyeong will realise the child is herself, because she still has the locket, and the child in photo does too. She will deduce that the lady in the picture is her mother. This is even if she does not remember her mother, cause its been long.

And of course Hideo will definitely recognise his adoptive mother cause he was relatively older. If Ga Kyeong reveals her own pendant, then he might show her his half, and connect basis having the same lady as one's biological mother and another's adoptive mother.

So Mrs Sato through the pendant becomes their thread of fate - making the two come together. And assuming the complete pendant came from somewhere - and how So Ran is seeing her daughter in Ga Kyeong so many times (After 13 years where before she was all quiet) - the assumption is that it was actual Soo-hee's before. So Ran also bought them (Hideo and Ga Kyeong) together even if it was the scheme + name Soo Hee.