
I think that the two girls doesn’t know that Taeha is an illegitimate son from an affair, We didn’t really get to know what really happened for that (if went on the news or things like that) so I’m just assuming that only the family knows the truth, raised Taeha like a legitimate child. Since Taehee knows the truth ever since he was a child, he didn’t hesitate to bully Taeha because of that..but I haven’t seen the father treating him differently because he’s an illegitimate son..on the contrary I think he has some favoritism towards him ! So in the end, from the public eyes, Taeha and Taehee are just two normal brothers from a Chaebol family (and it adds a LOT that they are literally copy + paste of their father)

If they don’t know, then that woman being the mother makes even less sense.
To the public, he’s supposedly a legitimate son, but he’s called illegitimate by his brother and the people working around them (like the security guards and bodyguards). Why would the security staff call someone illegitimate if he were the child of a legal marriage? The girls referred to the woman as the CEO’s "wife."
That raises several questions. If that woman were Taeha’s mother, why is Taehee so confident that he’ll inherit his father’s position in the company? Shouldn’t he be worried that the wife might pull strings to help her own son take the role?
As for favoritism, I think the CEO only favors those who are "useful." Jay himself said the CEO never truly cared about Taeha. The thing is, Taeha has been showing better behavior and performance than Taehee, and that’s the only reason he’s currently being favored — nothing more.
(Of course, I could be wrong and she might actually be Taeha’s mother. If that’s the case, there must be a very complex backstory behind the family dynamic to explain it all.)
May I be wrong but I don't think that woman was Taeha's mom.
Supposedly, Taeha is the result of an affair between the CEO and a mistress, which is why he's called a bastard. It wouldn't make sense for his mother to be married to the CEO — if that were the case, then where would Taehee's mother be? Taehee is the one supposedly considered the legitimate son, and it makes sense that he acts superior.
Think of it this way: it would be strange for Taehee to behave so confidently if the woman married to the CEO were Taeha's mother.
An alternative would be: Taehee's mother died and the CEO married another woman — but even so, it’s still strange, because if that’s the case, why do they keep calling him illegitimate?