I didn’t see an answer upon some cursory Googling. Is it an international convention?
I don’t know why “passengers” is shortened to “pax”. But I do know it’s part of a pattern where Wx means weather, Tx means transmission, Rx means receive (a transmission) and so on. So the bigger question is why words are abbreviated in this way in aviation-speak.
Maybe they picked it up from the medical world. Using x to abbreviate words is common in medical charts.
I write software for airlines.
From my perspective, it’s convenience. Do you know how many times in a given conversation/report/email/whatever you use the word “passengers”? It’s a long word that’s used a lot. It makes sense to shorten it, and “pax” gets the idea across.
Former airline employee.
It goes all the way back to the days when everything was sent via telex. Pax is a lot shorter than Passenger while still getting the idea across.
This is my own hunch with no supportive cite to add.
The trend is to abbreviate and acronym your way through “communication” even if no information is passed along. Gobbledygook for its own sake and amusement.
Where, I recall being told, the “x” is essentially equivalent a period. It indicates that the previous is an abbreviation, but unlike an actual period it doesn’t get lost with quick handwriting.
Because “PAS” would be pronounced “pazz”, which sounds lame.
You want to ask this over at Pprune.
Ironically, attempts to abbreviate the word communications itself were unsuccessful for reasons that were obvious in retrospect.
Much shorter to type on a Telex, or even into a computer. However, I’ve never come across it in railroad or maritime parlance. Especially in the early days, there was a lot of overlap between railroad and airline practice.
I see it used in the tourism industry in general. Ferries, buses, even the number of guests on a tour.
I was about to say that I regularly see “Communications” abbreviated as “Commo” or “Comm”, but then I got the joke. 
I’ve seen “Pax” used to described the capacity of ground vehicles as well, i.e.: An 8 Pax van, a 12 Pax van, a 3 Pax truck, etc.
I think mostly it’s just because it’s an abbreviation that can not easily be mistaken for any other word in context. I mean, it is also the Latin word for Peace, but that’s not likely to be taken as relevant to anyone discussing travel arrangements.