There are a few tip-offs that Verbal is Keyser Soze:
Before Kujan arrives, Verbal is shown glancing around the office, noticing details that he can turn into his impromptu story.
Before Verbal talks about Kobayashi, there's a shot of Verbal looking up at Kujan just as Kujan is taking a drink from his coffee mug, giving Verbal a clear view of the "Kobayashi" insignia on the bottom of the mug.
Verbal complains that he needs coffee or he'll get dehydrated. Coffee contains a large amount of caffeine, which actually accelerates dehydration. Verbal is faking his poor health.