Not quite, the requirement is that phones not bought locally need to be registered after 30 days, regardless if someone is a tourist or a Chilean citizen. It's a mix of deterring tax evasion, importing stolen phones, and regulatory homologation. The government delegates responsibility to various telecom companies to have portals to self-register the IMEI with ID, which can be a foreign passport.

I'm sure the cops can get that info, but its mostly to enforce the "only one free register per year". Anyone can buy a phone with cash and use a prepaid sim with zero ID needed.