That tea is - for the most part - known as "chai" in India suggests a different origin (the world's languages are largely divided on the first sound in the word for tea: either Te or Ch).

It is true that the British brought large scale tea production to India, but the plant is endemic to India and its local consumption long predates the British (though the exact origins are somewhat obscure).

https://en.m.wikipedia.org/wiki/Indian_tea_culture

It seems that words for tea that derived from "cha" spread across land, while the ones originating from "tea" developed over sea routes.

https://qz.com/1176962/map-how-the-word-tea-spread-over-land...

Same reason Italians love tomatoes and Thais love chili peppers I guess. Non-native ingredients seem to sometimes have a much greater influence in the new place than it’s original one

Historically, yes. But masala chai is distinctly Indian, and one of the more notable examples of a tea where milk is a core ingredient instead of an optional addition.

A guy I know in India once remarked that, say what you will about the British but one good thing they brought was milk in tea.