5

It's mentioned multiple times that Kazuto is Suguha's cousin, not her brother. However, the key conversation in question seems to imply that the explanation was made up on the spot. To me, it seemed as though that was best alternative Kazuto's mom could come up when Kazuto confronted her unexpectedly; in reality he's really adopted from unknown parents. I couldn't find a good clip of it, so here's the clip in a reaction video, which is close enough: https://youtu.be/X8DD3tMzrsg?t=1239

Earlier in s1 ep15 Suguha mentions that she and Kazuto look nothing alike. To me, that was included to specifically suggest that they're unrelated.

Admittedly I'm relying on the english dubbed version and I've never read the Manga, so I was hoping there was more in the original sources.

8

According to the light novels they are cousins.

Kirigaya Minetaka and Midori, husband and wife’s own child was Suguha, while Kazuto was Midori’s sister’s, this meant that he was the child of Suguha’s aunt. Her aunt and her husband, without a choice, left their only child, who was not even a year old, and passed away in an accident. The serious nature of the injuries they bared eventually putting an end to their lives, and Midori was left to take charge of Kazuto.
That fact was told to Suguha by both her parents during winter two years ago, around that time, Kazuto was trapped in a virtual reality game, going by the name of «Sword Art Online». Suguha who had already received a big shock from the incident, was in a cruel state of confusion; she went to Midori, asking why she did not tell her earlier and why she was the only one not being told, and took out her confusion on Midori.

A bit earlier it is stated that his parents were surprised when Kazuto asked after his real parents so I find it unlikely that they would have lied to him.

“... Even now, I recall that incident like it happened yesterday. That time when Minetaka-san and I were watching a movie in the living room, and Kazuto suddenly said from behind, ‘Please tell me about my real parents’ “.
Suguha motionlessly gazed at Midori’s charming lips with very light rouge, a light nostalgic bitter smile appearing on them.
“At that time, I was really surprised. Kazuto was only 10 years old. Till Suguha entered high school... it was a secret that we had planned to keep for 7 more years, but he found out about his Juki Net’s erase records.”

Both excerpts are from the prologue of the third light novel Fairy Dance (translations by Baka-Tsuki).

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