Grew up in India. As the sibling commenter notes, this is likely an older British expression that has survived in Indian English.

It's hard to remember now, but I think the way I thought about it when I was taught is that "3 × 4" is combining 3 and 4 into a larger number 12. The "3 ÷ 4" was not special, and "by" there was just an abbreviation of "divided by" like you say.

For what it's worth, I think times usually should work in Indian English as well, as in "3 times 4" for 3 × 4.