For example, imagine the name of our language was "Englishlanguage", and we had to always say "I speak Englishlanguage" or "She's a native Englishlanguage speaker". The closest I can think of is 日本語 "nihon-go" or the name for Japanese, where the "go" means "word", but this isn't exactly what I mean. I'm wondering if there is any language that contains the word for "language" or "tongue" or similar, in the native name for the language. For example, are there any cultures who name their language after themselves (e.g. "I speak LanguageoftheAngles but she speaks LanguageoftheFranks")
The (Equadorian) Quechua name for the language is Runa Shimi meaning "Language of the People". All of the Saami terms for the languages (e.g. North Saami, Lule Saami etc) include the word "language", such as Davvisámegiella. The Lushootseed name for the language, dxʷləšucid, means "language of the south" and contains a suffix -ucid which means "language" and "mouth" (there isn't really a separate word "language").
Yes, there are even endonyms like Fala that simply mean language/speech with no further qualification.
Tok Pisin, Papiamento, Lingua Franca etc are other examples.
(Tok in Tok Pisin is from English talk. Papia in Papiamento is from papear (to chat). lingua in Lingua Franca is from lingua (language, tongue).)
There is also the Turkic suffix -ce/-ca/-çe/-ça, which is used explicitly for languages but cannot stand alone.
Affectionately Yiddish is often referred to as מאַמע־לשון (mother tongue), if you search you can also find German dialects and languages referred to with Muttersprache, Mottersproch, Muddasprach, Muttersprak etc.
In Lithuanian, the name of a language is the genitve plural of its speaker + the word "language": eg. lietuvių kalba 'Lithuanian language(lit. language of the lithuanians)', anglų kalba 'English language', kinų kalba 'Chinese language'. However, the word "kalba" maybe omitted if the context allows.
In sentences like "I speak XXX language", the forms above are not used, they use an adverb instead. For instance, Aš kalbu lietuviškai 'I speak Lithuanian(lit. I speak in the lithuanians' way)'
An example from an Australian Aboriginal language is Murriny-patha, which might be glossed as language good.
Also referring to language families but not languages, we have examples like Langue d'oc vs Langue d'oïl (referring to the language groups' respective words for "yes").
Classical Nahuatl speakers refer to their language as Nahuatlahtolli, "Nahuatl language", from Nahuatl + tlahtolli. However they add tlahtolli to any other language, so "I speak English and German" is translated Nitlahtoa Inglatlahtolli ihuan Teutontlahtolli. In modern Nahuatl tlahtolli may be omitted: Nitlahtoa zan tepitzin Nahuatl, "I only speak a little Nahuatl".
The Sumerian language is known in Sumerian as eme-gir15, which means "native tongue" (Foxvog, Daniel A., Introduction to Sumerian Grammar. 2016 January 4, p. 21).
The Japanese language is known natively as 日本語 ("Nihongo"), which literally means "sun rise language", or, more loosely, "language of the rising sun".