Apparently, to those not all that familiar with it, allAnime and Manga are either geared towards children (like Pokémon) or Hentai. Which tends to be why fans of anime and manga are Acceptable Hobby Targets. Or more broadly, all anime, as opposed to "cartoons", which anime technically is, is for adults (and thus not appropriate for children), a stigma that exists among both anime fans and mainstream society. It's just the opposite, in fact. Because of cultural and societal differences, what Japan considers "family-friendly" is similar, but not entirely the same as the western definition. Often, those same people use the words "anime" and "manga" interchangeably, or not even realize that the word "manga" exists. (They're used interchangeably in Japan. Japanese people often use the word "manga" in much the same way English speaking people would use the word "cartoon", which refers to both comics, animation, or any other kind of drawing.)
Don't you know? All anime is sci-fi with 50 foot tall neural-interfacing robots. This is however sadly what a lot of professional US animators (such as is the case with John Kricfalusi ) seem to believe, mainly because Transformers was for a long time the only Japan-US collaboration.
Many people who aren't big anime fans have only heard of Pokémon, Yu-Gi-Oh!, Dragon Ball Z, Naruto, One Piece, and Sailor Moon. And most of these people believe that Dragon Ball Z refers to the entire Dragon Ball franchise, when it's really just the second of three series, after the original Dragon Ball but before Dragon Ball GT.
Speaking of Pokémon: Even though there are at least 150 basic Pokémon species, the great majority of non-anime fans have only heard of Pikachu (most likely because of his mascot status). Squirtle, Bulbasaur and Charmander might also be mentioned. It's also not widely known that Pokémon can actually mutate, and that Pikachu's "upgraded" form is Raichu, who is orange instead of yellow.
Some people think that Pikachu is actually called "Pokemon".
To a number of anime fans, all Japanese "anime" is AKIRA, Ghost in the Shell, Ranma ˝, or any other show, movie or OVA that has graphic violence, swearing (though that's more of an issue with dubbed anime; it's a long story), nudity, and/or other "adult content" that one (usually) wouldn't see in a western "cartoon". Never mind that the first animated film to be rated X was an American cartoonmade in the early-1960's by a man that used to work for a studio that had its influence on a lot of influential manga-writers who were young at the time, such as Osamu Tezuka.
Amongst Japanese viewers, Eiken (not the anime OVA, but the studio) tends to be known more for Sazae-san these days (it IS Japan's longest-running anime TV series). Back when they were called TCJ, they made once show that is still very well remembered in Japan to day: Tetsujin 28-go (AKA: Gigantor).
Speaking of Sazae-san, because the show is not very well known outside of Japan, some non-Japanese fans might think that One Piece is the longest running anime.note It premiered in 1999; 17 years ago in 2016, whereas Sazae-san has been running for 47 years, premiering in 1969.
Companion to the above: Many will think that Anime refers to the art style typically found in them and not realize that it is a catch-all term for Japanese animation and comic books. Expect anything Animesque to be called an Anime, regardless of whether or not it is actually produced in Japan.
According to the Franco-Belgian comic book publisher Dupuis the only manga worth remembering is Goldorak. Not even its original name (Grendizer) would be used.
If you're lucky enough to catch an anime reference in western media, chances are it'll be to Dragon Ball Z, Pokémon, or Sailor Moon by virtue of being massive phenomenon in the '90s. Gundam, Yu-Gi-Oh!note mainly in the '00s, and Naruto might also get a mention.
Anime and manga often make a Shout-Out to another anime or manga series. Sometimes fans catch the reference, because the series being referenced has also been released in the West, but sometimes it hasn't been, and is mistaken for a reference to something more familiar. For example:
Western anime fans are often unfamiliar with the delinquent "yankee" stereotype in Japan, and assume all delinquents are a reference to either Yusuke and Kuwabara or Jotaro, Josuke, and Okuyasu. The manga Otoko Ippiki Gaki Daisho stars a delinquent that may have inspired all of the above, and he's from a manga from the 1960's. The fact that the series hasn't made it to the States yet may contribute to its obscurity over here.
Motorsports anime and manga? To most people, Initial D is the only one that exists.
If a pre-80s anime gets referenced in western media, it's either Astro Boy or Speed Racer. Gigantor sometimes pops up as a third option. Don't expect a reference to any Astro Boy series besides the original, or to many characters besides Asto himself.
Aversions and notable exceptions of this trope from Anime and Manga:
Averted in Haruhi Suzumiya, mostly in the novels. We have a story which uses Euler's Planar Graph Formula as a plot device. Jean-Jacques Rousseau is mentioned in another short story extremely casually, and half the historical references are of Japanese history. Best of all, Yuki's books always refer to the current plot, like when she reads Hyperion in Melancholy. Koizumi, especially in Melancholy, peppers his words with philosophy, like the Anthropic Principle and the Omphalos Hypothesis. Even the title sequence for the first season isn't spared. Read up about it in the Genius Bonus and the Viewers Are Geniuses page.
Princess Tutu, an anime based around a Magical Girl Ballerina, smashed this trope. Classical music serves as almost all of the background music in the show, and while a number of famous works are included (for example, Pyotr Ilyich Tchaikovsky's The Nutcracker and Swan Lake), both more obscure composers (Smetana, Mussorgsky, Satie) and less-popular works from famous composers (Beethoven's Egmont Overture). And it features a lot of ballets, from Giselle to the aforementioned Cinderella to Coppelia.
Nodame Cantabile naturally also uses works not by Beethoven & Mozart. The animators love "Veni, creator spiritus" from Mahler's 8th, for example, a fact that escapes the Other Wiki's notice. And Purcell's Abedlazar...
The Gag Dub of Crayon Shin-chan includes references to many obscure things, all the way to making a reference to MOTHER. An interview by one of the writers said they deliberately tried to avoid this.
Hunter × Hunter features cameos and references to well known Japanese celebrities, but also much more obscure ones (one of the sadistic antagonists reading Trevor Brown probably takes the cake).
The fairy tale anthology anime Grimm's Fairy Tale Classics included many obscure fairy tales such as "The Iron Stove" and "Jorinde and Joringel", in addition to well-known ones like "Cinderella" and "Snow White."
Meta example: The Japanese surname "Yagami" is spelt with the kanji for "eight" and "god" — so, 八神, "eight gods"; it turns out to derive from a placename. However, most Western anime fans first encounter it through Light Yagami, who spells it with the kanji for "night god." This has resulted in at least two similarly-named characters on this very wiki being written up with incorrect name meanings of "night god." Whoops.
Don't forget the beautiful princess Yagami-hime (八上姫), in Japanese Mythology. She's spelt with the kanji for "eight" and "rising up".