Americans Hate Tingle: Barbie is simply disliked by a lot of Japanese fans, due to her grown-up nature, compared to the 11-year old Japanese doll, Licca-chan, but in America? She still goes toe-to-toe with Mattel to this day. She's fared slightly better as a fashion label over there, but not by much.
1993's infamous "Earring Magic Ken" was discontinued due to accidentally looking too much like a Camp Gay stereotype. It was also the best selling Ken doll in the franchise's history specifically because gay men brought so many of him.
There's also a niche following of gay/bi women who grew up with the dolls.
Depicting Barbie as a Lipstick Lesbian isn't unknown amongst fans. She usually gets paired up with her buddy Teresa. Some of the direct-to-video films make them easily shippable as well.
The Problem with Licensed Games: They are made for little girls, and almost none try to reach beyond that. Especially the ones with interesting concepts, you'd think a Barbie designs stuff game would manage to be entertaining. Though some of the older ones have become minor So Bad, It's GoodCult Classics.
A lot of collectors, especially older collectors, liked Barbie better when she had her original "cat eyes".
Fans of the films act this way when Barbie's speaking voice isn't Kelly Sheridan. She was replaced with Diana Kaarina for A Fashion Fairytale, brought back for Barbie in A Mermaid Tale 2, and then replaced again later with Erica Lindbeck.
In the 2000s, Ken and Barbie broke up and Barbie gained a new love interest. Fans hated that Mattel broke them up. However, it was just a short-lived publicity stunt.
Fans of the films have had mixed reception towards the 2010s changes to the series. Namely, usually portraying Barbie as a teenager instead of an adult (complete with a more youthful voice).
The 2013 Barbie book I Can Be a Computer Engineer was accused of teaching girls not to pursue technological careers, but rather that they should always get men to solve their problems, then take credit for their work. Amazon and Mattel respectively responded to the controversy by ceasing all direct sales of the book, and promising to write stories with an "empowered" Barbie.
Infamously in the 90s, "Teen Talk Barbie" caused this due to its controversial line involving math. The actual line is "Math Class Is Tough!". Mattel's intentions were relatabilty - few school kids like math and high school math classes are often considered hard - but parents heard it and were offended. The line either perpetuated Dumb Blonde stereotypes or made it seem like all women were bad at math. A good chunk of this issue ended up being due to the line being remembered incorrectly. It's often said that Barbie's line was "Math is hard!", or something similar, which sounds more like it perpetuates the idea that "girls are dumb". The doll ended up recalled as a result.
In the early 2000s, Mattel introduced the "Happy Families" line. It was discontinued due to complaints that it was promoting Teen Pregnancy. Midge and Alan were actually Happily Married and in their 20s at the time, however many people associate Barbie dolls with teenagers due to Barbie's fluctuating age.
Values Dissonance: Among other things, Barbie's and Ken's first commercials each end with Barbie getting married.