How is Lupita Nyong'o the first Mexican actress to win an Oscar?

Because, she's a Mexican National by birth under Jus Soli, the common law concept of Nationality by soil, enshrined in Article 30 of the Mexican Constitution.

From Wiki:

Nyong'o was born in Mexico City, Mexico, to Kenyan parents, Dorothy and Peter Anyang' Nyong'o, a college professor turned politician. Nyong'o identifies as Kenyan-Mexican. It is a Luo tradition to name a child after the events of the day, so her parents gave her a Spanish name, Lupita (a diminutive of Guadalupe).

She also studied in Mexico for a year when she was 16.

Based on a few interviews, and because she doesn't seem to have ever resided in Mexico when she turned 18 or after, it doesn't seem she has Mexican Citizenship, which is separate from Mexican Nationality. In interviews, she always refers to nacionalidad instead of ciudadanía. In the US, US nationality is also separate from citizenship, but this only applies to very very few people (A person whose only connection to the U.S. is through birth in an outlying possession, currently just American Samoa and Swains Island).

She has never given up or renounced her Mexican nationality, and Mexico recognizes dual nationality/citizenship, even after birth. In fact, she couldn't renounce her Mexican nationality even if she wanted to.

So because she is a Mexican National by birth, the claim of her being the first Mexican Oscar winner, by her or others, is legit. There has been no other Mexican woman to have won an Oscar, as actress or other roles, with only 4 ever nominated.

• 1954 Katy Jurado Broken Lance Nominated First Mexican-born and Latin/Hispanic actress to be nominated for an Academy Award
• 2002 Salma Hayek Frida Nominated First (and only) Mexican-born actress to be nominated for Best Actress & the second Mexican-Born actress to be nominated for an Academy Award
• 2006 Adriana Barraza Babel Nominated Third Mexican-born actress to be nominated for Academy Award
• 2013 Lupita Nyong'o 12 Years a Slave Won Mexican-born Kenyan actress. First Mexican-born actress to win an Academy Award.

There have been Mexican men to win Acting and Directing Oscars. A comprehensive list on Wiki.

She self-identifies as Kenyan/Mexican, due to having been born in the country (to Kenyan parents) and having lived there for half a year after her 16th birthday.

“I was born in Mexico because my father was teaching at a school in Mexico City,” the 30 year-old actress explained. “I was born during the third year he was there. And when I was 16, I returned to Mexico to learn Spanish. I lived in Taxco, Guerrero and attended the UNAM, the Education Center for Foreigners. [I] lived there for seven months and I loved it. ”

“Mexico was very strange to me, coming from Kenya, but I found such friendly people and fell in love with the food,” the Oscar-nominated actress told CNN en Español. “To this day, I love eating steak tacos before going to the red carpets.” - Huffpo interview

and yes, she's aware of the controversy her nationality is causing

“I’m Mexican and Kenyan at the same time,” she said, according to El Mañana. “I’ve seen the quarrels over my nationality, but I’m Kenyan and Mexican at the same time. So again, I am Mexican-Kenyan and I am fascinated by carne asada tacos.”

She talks about her upbringing and her love of Mexico in this cast interview for 12 years a Slave