The dictionary of etymology says that the word sigh dates to the 1200s and is probably echoic of the sound of the act of sighing.
The name of the Bridge of Sighs in Venice was said by Lord Byron to originate in the sigh given by convicts seeing their last view of Venice as they crossed the bridge to their summary execution in the time of the inquisition.
But in fact not, because by the time the bridge was built the inquisition was long over and there were no more summary executions.
The Bridge of Sighs in Cambridge is a covered bridge at St John’s College, Cambridge University. It was built in 1831 and crosses the River Cam between the college’s Third Court and New Court.
It has nothing in common architecturally with that in Venice, save that both are covered and so the one echoes the other.