Essentially, pax means people/persons/occupants. as succinctly expressed by Callithumpian's answer (apparently it was used as early as the 40s; it became a standard term in the UK Passenger Transport industry in the 70s).
I worked in the bus industry for many years. Pax isn't exactly shorthand for Passengers. It's short for Passengers and Passes. In which context, Passengers is short for Paying Passengers .
The reason being that from a bus company's point of view it's crucial to distinguish between Passengers (who pay the fare when boarding), and Passes (using a Season Ticket, or some other pre-paid authorisation to travel).
In the UK, Local Authorities routinely pay for passes used by pensioners, for example. They may also pay a bus company a flat sum simply to run a Service that the company would otherwise have considered uneconomic. In such a context, the bus company needs to analyse Pax totals to ensure their buses aren't being overloaded, and Pass totals because the Local Authority wants to know their money was well spent. And they need (Paid) Passenger totals because those are the real 'customers' who might use a different bus company if they don't like the service or the price.
It may help to point out that one of the reasons for using Pax is simply to make report layouts easier. I
was coding up those reports in the 70's, when things were a bit more constrained than today. Managers couldn't just run off a report when they wanted it; they got standard reports run overnight by computer operators.
These were often very long printouts, so page space was at a premium and columns were as narrow as could still fit the data in. Instead of a single column showing just 'traveller' totals, managers obviously wanted the breakdown. So we gave them Psg Pas Pax. being Paying Passengers, Pass Users, and Total Travellers.
Finally I've never actually seen pax used outside the transport industry, but it's worth pointing out that a dog. for example, probably wouldn't count as a pax. even if a fare was payable. That's because a dog doesn't occupy a seat. By the same token, a baby in a portable cot probably wouldn't count as a pax in an apartment, because it wouldn't occupy 'bed space'.
EDIT: As several people have commented, the usage has been extended - particularly in the Far East, and most particularly in the hospitality industry. So whereas originally pax was always travellers (live human bodies that need to be transported) it's often now more generally applied to any "customers, people, bodies" occupying space (usually, seats or beds ), who must be entertained, accommodated, fed, etc.