The words "anime" and "manga" both originate from Japanese, and have no English roots (though "anime" comes from French). To me, it doesn't make sense for an "s" to be attached ("animes" and "mangas"). However, I still see people do this quite frequently, so it's hard to tell for sure.
What is the correct way to pluralize the words "anime" and "manga"?
As far as the "English" language is concerned, from the Wiktionary entry about "anime":
Noun: anime (countable and uncountable; plural anime or (proscribed) animes)
(uncountable) An artistic style heavily used in, and associated with, Japanese animation, and that has also been adopted by a comparatively low number of animated works from other countries
I can draw an anime version of you, if you want.
(countable) An animated work originated in Japan, regardless of the artistic style.
2005, Peter J. Katzenstein, A World of Regions, page 165,
After three months of successful sales in manga form, it was made into an anime for television.
2005, Joan D. Vinge, in The Year's Best Fantasy and Horror: Eighteenth Annual Collection, page cix,
Usually the manga comes first, though it may be an offshoot of a novel, and an anime may be inspired by a video game.
2006, Thomas LaMarre, in Japan After Japan (Tomiko Yoda & Harry D. Harootunian, eds.), page 363,
These anime prepared the way for Otaku no video, a two-part Original Video Animation (OVA).
(rare, countable, chiefly proscribed) An animated work in anime style, regardless of the country of origin.
Likewise, from the Wiktionary entry about "manga", it's handled a bit differently:
Noun: manga (countable and uncountable; plural manga or mangas)
Lately I've been reading a Brazilian manga.
(Note that 2 of the examples from the countable plural of "anime" also apply to "manga")
So in the case of anime, the non-proscribed (or exclusionary) plural is unchanged, but the rarer proscribed plural can be used with an "s". The case with manga seems to be either way as far as Wiktionary is concerned.
The Collins Dictionary, the New World Dictionary, Oxford Dictionary all say that manga is also the plural form. Horever, the Macmillan Dictionary says that mangas is the plural form (need to click on "Word Forms").
If you're using it as a Japanese borrowed word (either "manga" or "anime"), it isn't incorrect to use the same form as singular and plural, like the Japanese words themselves.
They would probably be "anime" and "manga," just like the plural of Pokémon is Pokémon.
Most nouns in Japanese have no plual form, so you use the same word whether you're using a singular noun or a plural noun.
There are rare exceptions when a noun does have a plural form, but usually the plural form simply repeats the word using a "repeater" sound for the first consonant.
E.g. hito (人 [ひと], person) becomes hitobito (人々 [ひとびと], people) and kami (神 [かみ], god) would be kamigami (神々 [かみがみ], gods).
It depends.
If you want to go based off of Japanese, then the plural would just be "anime" and "manga". There is no normal way to pluralize these words in Japanese; rather, you would say something similar to "anime shows" instead of specifically multiple anime.
My #1 favorite anime is Pokemon.
My two favorite anime are Pokemon and Digimon.
If you want to go based off of English, then the plural would be "animes" and "mangas". This is how most pluralization occurs in English.
My #1 favorite anime is Pokemon.
My two favorite animes are Boku no Pico and Digimon.
Even though they are Japanese words, if you are using them in English sentences, it's not all that weird to pluralize them like an English word. Café comes from the Romance languages, but would you be all that irked if someone said "There are two cafés on that street"?
It's really a preference. Your message will probably be properly conveyed either way. Whichever one you pick, someone will think it's wrong, so you might as well go with the one you prefer. I've seen both pluralization methods used quite often, so I wouldn't be greatly shocked by either.
It's mainly just preference. Since the Japanese language does not have a way of pluralizing that applies to most of their words (like adding an 's' to the end of words in English), it is usually left to preferences.
Some people like to say 'manga' and 'anime' as the plural form of the words since it sounds more "correct" and is similar to the way the plural version of 'deer' is 'deer'.
Other people prefer to say 'mangas' and 'animes' because in English it is common to add 's' to the end of words to pluralize them.
Since there is no definite way of pluralizing these words, you can just pick whether or not you want to say the words with a 's' or not.
