Yes. Also, no.
“The Chinese language” can really only refer to the written form, which is the same all over China.
Mandarin is a Chinese language. Many languages are spoken in China; the ones related to Mandarin are collectively known as Hanyu. They can be grouped in 7–12 different branches; the best-known in Western countries is probably the Yue branch, to which Cantonese belongs.
Those branches are not really mutually intelligible; linguistically, they are separate languages. (This is why reforms of the written language have been so difficult: since they don’t use letters but ideograms, the same written text can be used all over China. Calling the Chinese languages “dialects” is on some level more of a political statement of unity.)
You can compare it to the Germanic languages: some are very close, like Danish and Norwegian; others are more distant, like English and German. And while you can talk about “Germanic languages” as a unit in some contexts, it doesn’t really make much sense to speak of them all as “English” just because it happens to be more widely spoken. In the same way, it’s not correct to call all Chinese languages “Mandarin” just because it happens to be dominant, comprising some 2/3 of all speakers of Chinese languages.
Of course, if you pick a Chinese person at random, there’s a good chance that the language they speak is Mandarin. And some 90% of the population is more or less fluent in Mandarin. But you can’t properly understand the plot of Everything Everywhere All At Once unless you understand that Evelyn’s family are from Hong Kong and speak Cantonese, while Waymond speaks Mandarin; they met in Hong Kong where Waymond was a refugee from China, of much lower social standing. And it’s a plot point that their daughter speaks Cantonese, but struggles with Mandarin.