Have you considered that he could have been from the former Kingdom of Lombardy-Venetia? This area was part of the Austrian Empire to Italy’s north, whereas Trieste was more northeast.
Civil registration in northern Italy apparently began in 1866, probably after the territory became part of the Italian kingdom. Prior that I would expect practices similar to the rest of the Austrian Empire, which means the churches.
Church records which are presently in Italy have limited accessibility, all that is evolving. Most, with some exceptions, are not at FamilySearch.
That said, unless you have something that puts your ancestor in a specific town, chances of finding him are not high.