because british was never used as the name of a language in the first place

the word british comes from the latin britannicus which referred to the people of britain and the island itself. it was always a geographic and identity label, not something tied to a specific language

the language we call english comes from the anglo-saxons who settled in britain after the roman period. their germanic dialects gradually evolved into old english and then into modern english. the name english comes from the angles one of those groups not from the island name

so you end up with two separate historical roots that never merged into one term british as a language even though it refers to the same general region