1. Anime has not been around long enough
The first anime was created in 1907, but the first produced and televised anime didn't came until 1961, which is more than 50 years after the first anime . The first light novel with anime-ish style came in 1970s . The first modern manga originated in 1945 . It hasn't been a century before anime and manga exist.
In comparison, many of the Western Classics came in the 19th century, for example, The Prince and the Pauper (1881), Wuthering Heights (1847), Pride and Prejudice (1813), A Tale of Two Cities (1859). Some, such as Shakespeare's came from even further back in the 16th century. This form of art has been long enough that people have long enough time to know them and start critically analyzing them, which also contributes to my second point. It has been read by generations of human population.
Most people who went to school would have at the very least heard of Shakespeare's Romeo and Juliet, western or not. The same cannot be said about anime. Only avid anime and manga fans would know of Fushimi Tsukasa's Ore no Imouto ga Konnani Kawaii wake ga nai. Even with world-wide popular anime such as Naruto, only some would notice the name Masashi Kishimoto. Anime and manga has yet to reach the popularity that classic literature enjoys.
3. Content of anime
Being made in Japan, many if not most of anime and manga have a view on God that would be regarded as insulting to those living in religious countries. I remember when I was younger, I turned on TV to find Saint Seiya broadcasting and there one of the characters said something about "power equal to that of God". My father overheard that and told me to change the channel ASAP. This is due to the way Japan perceive God, that in Japan there are many gods, so much that it was said that Yamato (ancient name of Japan) is a land of thousands gods. Older people tends to be more religious as shown in the statistics, more people in the 80s are religious compare to the 90s and most of them are a follower of Abrahamic religions which is monotheistic and regards polytheism as sinful .
Another problem with the content of anime and manga, especially in modern anime and manga is that many of them contains fans service that is sexual in nature. While pornography is legal in western world, in the other parts of the world it is illegal. China and Indonesia, two of the most populous countries in the world, that is approximately 20% of the world population, have pornography as something illegal. From these two countries alone we already lost 20% of potential critical reviewer.
4. You're an otaku, pffft
When someone is known as an anime and manga lover, he will get the label, otaku. This is especially true with adults. The problem with this is that otaku is not seen in positive light. Even in Japan otaku was seen in negative light and only in 2013 that a research shows it has been seen in more positive light. When you make a critical analysis of anime or manga, then like it or not you have to read or watch it. Most kids won't be able to make make a critical review, which means most critical review would come from adult, and like I said adult watching anime was not regarded in positive light. Like shown in many anime, announcing yourself as otaku is basically a social suicide, especially when many have questionable content.
5. Era difference, the Internet, and forms of entertainment
During the era of Shakespeare and other western literary classics, there was not many form of entertainment. The wasn't video games, there wasn't Internet, no device where you can listen to music, heck, even electric light only gets famous in the 20th century. This lack of entertainment means makes theatrical plays and books has little to no contender. Today with the variety of entertainment means available, less attention is spend on each. This also translate into less of the world population watch anime and read manga and light novel compared to when the previous era, where means of entertainment are limited. Had anime and manga was born during the Shakespeare era and Shakespeare during anime and manga era then the popularity would be reversed, with anime and manga receiving more critical reviews than Hamlet, and Romeo and Juliet.