You asked this question ages ago but I was curious too lol
It’s just a mistake, both names should be the same, ‘Kazuki’.
His brother’s name uses the kanji for moon 月, which is usually read as ‘tsuki’, but the author specifies that when it’s used in his name, it should be read as ‘zuki’, which the translator obviously missed.
When ‘Kazuki’ is used for Haruhi’s fake name, the author doesn’t write it in kanji, but writes ‘カヅキ’ instead. This is super easy to read, no way to misspell it, so the translator gets it right every time it’s written out like that.
Both ways are said exactly the same, ‘Kazuki’. The different writing is just to make it easier to differentiate if they’re talking about Haruhi’s fake name or his brother’s real name.
Haruhi goes by KaZuki when sleeping around and outright says, I think in the last two chapters, that he was using that name to defile himself and his brother. But his brothers name is KaTSuki...
Katsuki and Kazuki are definitely not the same name, and they never refer to the brother as Kazuki. Nor so Haruhi's sex partners ever use Katsuki, so it's not a mistranslation as far as I can tell.
Is this a lost in translation thing where if I understood the original Japanese itd be like a same Kanji, different pronunciation deal?