I aways thought that "Fat" Clemenza was his nickname, I always read that Fat was his nickname
On the FBI's Family Organization Chart, Peter Clemenza's nickname is Fat Clemenza. Sollozzo's use of the nickname could have just been a way to show Tom that he was familiar with the street names and habits of the Corleone family. In a small way a show of strength - knowledge is power.
Interesting that Sollozzo knew Fredo was a non-starter, and assumed Michael was a "civilian," because when he refers to them he mentions them merely as the other kids. I assume Connie did not count because women were not directly involved in mob operations back in the day.
"Io sono stanco, sono imbigliato, and I wan't everyone here to know, there ain't gonna be no trouble from me..Don Corleone..Cicc' a port!"