I've always thought that the name of this card game comes from the English word bridge (the structure) but it is not quite like that. It's the English pronunciation of a game called Biritch, which was also known as Russian Whist.
Etymology of the word from Etymonline:
card game, 1886 (perhaps as early as 1843), an alteration of biritch, but the source and meaning of that are obscure. "Probably of Levantine origin, since some form of the game appears to have been long known in the Near East" [OED]. One guess is that it represents Turkish *bir-üç "one-three," because one hand is exposed and three are concealed. The game also was known early as Russian whist (attested in English from 1839).
Furthermore from the book "The Theory of Gambling and Statistical Logic" By Richard A. Epstein:
This is the furthest I can go in my research (looks like an answer already) but is it possible to go further? Even it says obscure, are there any sources that goes deeper and gives more details (for the following questions especially)?
Why is the word corrupted to a word — bridge — that is already used? For example, why not britch? Is there a folk etymology here?
How come there is a Levantine and Turkish origin? Can it be that Russians learned this game from Ottomans in the era of wars, or did they adopt a name from the Ottoman Turkish language? (Also, the passage from the source book mentions that Biritch is not a recognized Russian word.)
Note: "Biritch" does not sound like a mispronunciation of some Turkish word or phrase meaning no trump. Interestingly, a Turkish language etymology website says that the name of this game briç is a loan word from English bridge. But this is about the modern Turkish language; there is also the Ottoman Turkish language used at that time which I can't trace back.