Why is the IATA Airline Accounting Code abbreviated as PAX?
It is called PAX because this is the accounting code for passenger operations. IATA has two kinds of these codes called Accounting Code for passenger operations and Prefix Code for cargo operations.

Accounting or Prefix codes
Airline accounting codes and airline prefixes are essential for the identification of passenger and cargo traffic documents, processing of passenger accounting transactions, cargo transactions and other commercial/traffic purposes.

(iata.org)
An airline can have both codes if they are operating both passenger and cargo flights, as seen in the following list of requirements to obtain such codes:

For a 3 Numeric Accounting Code (Passenger Operations)

Valid Air Operating Certificate (AOC).
Certificate of incorporation / Company Registration, with shareholders name.
Confirmation issued by the data aggregator (i.e. OAG, Innovata)

(only if a scheduled airline)

One of the following criteria must be meet;

Passenger Bilateral Interline Agreement with an IATA code holding airline
Confirmation from BSP (IATA Billing and Settlement Plan)/ARC (Airline Reporting Corporation)



For a 3 Numeric Airline Prefix Code (Cargo Operations)

Valid Air Operating Certificate (AOC).
Certificate of incorporation / Company Registration, with shareholder names.
Satisfy requirements to publish schedules in airline guide publications (only if a scheduled airline)
One of the following criteria must be met;

Cargo Bilateral Interline Agreement with an IATA code holding airline.
Confirmation from CASS (IATA Cargo Accounts Settlement System)
Letter of confirmation from a Freight Forwarder that will be issuing AWBs on your behalf



If applying for both codes, both requirements for each code will need to be met.

(portal.iata.org)
In your example, Cathay Pacific has both an Accounting Code and a Prefix Code of 160 because they operate both passenger and cargo flights.