At least that's how it always in American English

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To me, all of ()[]{} (and depending on context <>) are “brackets” in the sense that they bracket whatever is between them. Then a brace is a single side of a bracket (ie. opening brace plus closing brace make a bracket).

So to distinguish between the different brackets/braces you use “round”, “square”, “curly” or “angled” respectively. And “round” is the default so you often skip specifying that.

That’s been the convention in programming and maths everywhere I’ve worked outside America (in English).