Zack/Cloud *unmarked spoilers*

[TheAbominatrix]

A combination of stress, guilt, and inferiority caused Cloud to take many of Zack's memories and make them his own; memories he learned from long conversations with Zack. After Zack was shot and killed by Shin Ra soldiers just outside of Midgar, Cloud is found by Tifa (in Zack's clothing no less) who he wanted desperatly to impress (especially when he was younger) so he began to use Zack's life, saying he was in SOLDIER and all that (this is just one of the catalysts, mind you, but it seems to be the first time Cloud does this so it merits mention).

He doesnt directly have Zack's memories, only what Zack told him, as proven by the fact that Cloud doesnt 'remember' Aerith, who had been Zack's girlfriend.

edit: Oh, and to be clear, Cloud believes every word he says the truth. So all this stuff about being in SOLDIER and whatnot, he isnt conciously lying everytime. Tifa knows something is up, so she begins to prod him gently, reminding him of things and trying to get a clear picture of what's wrong. She only knows the extent (and can only help him out of his false identity) when she's in his mind in the Lifestream.