Explanation here:
southmk2
#xy 164 40
{CLOUD}
“It’s been a while,Mr President.”
------------------------------
#xyh 8 64 2
President Shinra
“Been a while?
Oh,it’s you.”{NEW}
“The Soldier who joined
this Avalanche,or whatever.”{NEW}
“Only those exposed to Mako
have that glow in their eyes.”{NEW}
“Tell me,traitor,
what is your name?”
------------------------------
#xy 200 56
{CLOUD}
“It’s… {CLOUD}.”
------------------------------
#xy 8 64
President Shinra
“You’ll have to forgive me.
I can’t be expected to
remember every Soldier’s name.”{NEW}
“You would have to be
another Sephiroth.”
It was brought to my attention by White Wind that it's unlikely Cloud and Shinra would know each other - or that Shinra would even know Cloud was a soldier.
With much discussion with Luksy too, it seems this is the explanation:
Very unlikely: Shinra has heard reports of a man wearing a 1st class Soldier uniform.
Unlikely: Shinra thinks that Cloud is Zax... a Soldier who disappeared. Although he was shot dead so...
Likely: The writers created a plot hole because they wanted to shoe-horn in exposition. Cloud thinks he has met Shinra before simply because the elite of Soldier will have met the president at some point (ceremony for new 1st class soldiers, for example).
The above (slightly corrected) is as good as it gets.