"Customer" has the same connotations in US English to native speakers. You aren't missing anything, and nobody would call their friends "customers."

But the software is using "CRM" as a term with its own meaning. It's kind of like "MP3" - the M and P come from "MPEG," where they stand for "motion picture." But really, "MP3" means "audio format 3 from the MPEG standard." Nobody thinks that MP3 users are calling audio files "motion pictures."