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)
- (uncountable) An artistic style heavily used in, and associated with, Japanese comics, and that has also been adopted by a comparatively low number of comics from other countries.
- (countable) A comic originating in Japan, regardless of the artistic style.
- (rare, countable, chiefly proscribed by fandom slang) A comic in manga style, regardless of the country of origin.
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.