I agree that his memory is knackered; the issue isn't that. The issue is that he thinks he's on first name terms with the president - which he would not. There's nothing leading up to it and nothing after it that allows for that jump in logic. As I say, even if you factor in that he thinks his life is Zax's, it does not make any sense. I remember a few times thinking "huh" at the scene. I even took this to Luksy, the Japanese translator for the retranslation project, because I was sure it had to be some sort of mistake. The very fact I did that—and that this thread is here—should prove the exposition is bad - even if one can find an excuse for it. I just don't buy it. The writers can get away with it because we can just say "Yup. Cloud is bonkers." but it doesn't really satisfy scrutiny. It's a small thing anyway - there are far bigger issues with FF7s story as a whole, but, thankfully, it's mostly a well-written and clever fiction.
The OPs question is explainable - President Shinra has heard reports about a rogue ex-Soldier joining Avalanche and now he sees Cloud with blue Mako eyes talking about a time they met. My issue with this exposition is a different and worse one.