Yes, and not only in anime. The stereotyping of language by gender, occupation, social class, etc. is pervasive in Japanese popular culture. The linguist Satoshi Kinsui calls this practice yakuwarigo [1].
It also exists in translations into Japanese from English and other languages, with dialogue in novels being rendered into stereotyped forms, even though the original dialogue was not marked for gender and Japanese people often do not speak that way in real life, either. I worked as a Japanese-to-English translator for many years, and those gendered translations in the other direction always felt strange to me.
I’ve discussed this issue with Japanese translators several times over the years. Most of them said that they were uncomfortable about using those stereotyped forms but that their publishers and, they presumed, the readers expected it. Times are changing, though. Just last month, the Asahi Shimbun ran a series of articles [2, 3, 4; paywalled and in Japanese] about changing attitudes toward gender stereotyping in translations into Japanese.
In real language use, sociolinguistic differences often vary by the individual and the situation. I’ve known Japanese women who never—except maybe when being sarcastic—use the pronouns or sentence-ending forms that are typical of female characters in manga and anime. The faculty meetings at the university where I now teach are conducted in very formal Japanese, and I have never noticed anyone use an expression in those meetings that seemed typically masculine or feminine. Real-life professors don’t talk like professors in manga, either.
[1] https://en.wikipedia.org/wiki/Yakuwarigo
[2] https://digital.asahi.com/articles/ASPCB6KD1PBTUPQJ009.html
[3] https://digital.asahi.com/articles/DA3S15110092.html
[4] https://digital.asahi.com/articles/ASPCD3FPHPBPUPQJ014.html
Japanese does have gendered first-person pronouns, though the textbook standard one "watashi" is neutral. In Japanese, pronouns in general (including fp) are used less than English; they are only used when there is ambiguity, names are often preferred to pronouns, and in casual speech it is even OK to refer to yourself by your name (like Jimmy from Seinfeld.)
In the textbook, Japanese doesn't really have gendered patterns to verb conjugations. Verbs are rather conjugated based on level of formality, for example "itte-ru", "itte-imasu", "itte-orimasu" all mean "I am going" in ascending degree of politeness. There are also impolite/brash-sounding forms "iku-zo", "ikou-ze", "ike", etc. which tend to be said more by men (or teenage boys) but women can use them too, it's just not "lady-like".
That said--it is fair to observe that in film for example samurai and geishas speak essentially two different sounding languages, more so than a knight versus a princess. Men are expected to be rugged and use "rugged" words; women are expected to be beautiful and use "beautiful" words. There are definitely words, speech, and tonal patterns which are more masculine and feminine, but in anime for example these differences are exaggerated far beyond real-world conversational norms.