Our male lead intentionally renouncing his very powerful 'adopted' identity and giving no name for himself—except that of Hong Heejoo's husband—has HUGE implications in Korean culture.
Due to their deep-seated Confucian beliefs throughout their history, women were rarely known by their actual names both in public and in private. They were know only by names of the men in their lives (this man's daughter, this man's wife, this man's mother) or by the title of their status or occupation. That still exists today to a certain extent. Also, women were very restricted on when and where they could go places publicly and had to hide their faces when they went out. The individual identity of women was therefore hidden from the world since they were not allowed to be truly seen or spoken about in a way where their unique defining personal attributes could be known.
So when our male lead denies his adopted name and powerful family status and then proceeds to publicly proclaim all his wife's personal attributes that uniquely define her—including a large picture of her face—while only defining himself as her husband, he flipped the traditional Confucian gender roles on their head.