Zack didn't die in Nibelheim; after Cloud tossed Sephiroth into the Lifestream beneath the reactor, Hojo and company showed up and took the incapacitated two of them to the laboratory beneath the Shin-Ra Mansion, where he experimented on them for a few years. They eventually escaped and made their way to Midgar, during which time Cloud was seriously out of it. When they finally got near Midgar, Shin-Ra soldiers caught up with them and shot Zack to death. Cloud freaked out even more, took Zack's Buster Sword and SOLDIER uniform, and reached Midgar by himself; by the time he met Tifa at the Sector 7 train station, he was gaining coherence but had unconsciously replaced some of his key memories with those of Zack. Things like having been in SOLDIER and wanting to be a mercenary were things Cloud either saw Zack do or heard Zack talk about.