Is קֻדְשָׁא בְּרִיךְ הוּא a translation of הקדוש ברוך הוא? At any rate both expressions are one phrase. Why then in kaddish do we reply with בריך הוא when someone says שְׁמֵהּ דְּקֻדְשָׁא, בְּרִיךְ הוּא or שְׁמֵהּ דְּקֻדְשָׁא. This question is related and Double AA in a comment quotes Taz OC 56 sk 3 who clearly reports בריך הוא as a response.
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The [Magen Avraham] says that the custom is to say brich hu with the Chazan so that he wouldn't stop there (like the Rama requires and unlike the Mechaber who says that one should stop there).
In other words, one is saying (in Aramaic) "...should be praised the name of The Holy One Blessed is He who is above all blessings, praise, etc."
Now, people made a mistake and interupt the Chazan with "Brich hu" which goes against the Rama.