Your are issued a diver (actually certification) number and it is indeed preserved but since the recreational cards are processed automatically, it depends on certain factors:
(1) your name and date of birth is entered exactly the same as prior certifications. Often a middle initial is missed in which case a new number is issued
(2) you were cerrtified in the same regional office - PADI have offices that cover specific regions and they act autonomously (PADI Asiap Pacific, PADI Americas, PADI Japan, PADI International etc). If you did one cert in the Caribbean and another in Thailand you won't get the same number.
(3) You can reconcile all your diver numbers and if you wish, get your cards re-issued to reflect that (it's not a requirement)
It has to be said that the diver number is not necessarily relevant because the data that is held by PADI can be searched on name and date of birth. If we for example, wish to check a diver's history, we can do so with a simple database search. Differences in names do occur, but mostly I've been able to resolve queries about divers who have forgotten their C card or logbook.
Remember that PADI divers number in the millions and there are about (just less than) 160,000 active PADI pros in the industry (according to the latest stats). The card is proof of certification, the diver number is just a means of tracking paperwork.
A PADI professional has a more immediate requirement that their certification number is readily available which is why we are issued with the pro numbers - these are unique and easily communicated between regions.
So the long and short of it is that there's no real need for a person to have a diver "number". One might argue that a credit card system which logs certifications and number of dives might be something to look at and 20 years from now, I'm sure it will. Right now, it's too much of a logistical headache. Remember some dive centres still get their mail processed by donkey and weekly boat plane (seriously).
And... really - do you want to be just another number!?
Happy diving
C.