It's slightly different in Australia. Here, instead of every citizen having a Social Security number, we each have a tax file number (if we're over 15, or have managed to earn money professionally while younger - difficult child 3 got paid for being in "The Black Balloon" and because tax got withheld, the tax office allocated him a tax file number). We also each have a Medicare number, from the moment we're born. At first it's the family Medicare number, with the last number indicating our place in the family order, usually by age. As our kids have gotten older and more independent, they have applied for their own card (and hence their own Medicare numbers).
For us, the Medicare number is needed in order to claim part or all of a doctor's bill, back from the government health system.
The Medicare numbers are also used to track general health system usage including purchase of restricted pharmaceuticals. It's law to show the card to a pharmacist who requests it. Whenever we go to a pharmacist we've not been to before, we MUST show our Medicare card or we won't be sold any prescription medication. This also applies to the restricted OTC medications, such as pseudoephedrine and asthma medications (plus a few others which the government has flagged as needing a pharmacist's counselling).
Tax file numbers MUST be provided to employers who are withholding tax or you are taxed at the maximum rate of 45%. You can claim it back, but to do this you need to send in a tax return - with a tax file number. They're easy to get, though.
We have other numbers if we're in business - it relates to our GST (Goods and Services Tax) which has been a goldmine for our government. The GST is linked in to each of us individually by our Medicare numbers, our tax file numbers and our ABN if we have one (Australian Business Number).
The system is legislated and computerised so tightly that you can't easily rip off the system. It also means the government can track your income, your business and your health records.
Social security - that's another (less important) number. It relates to any government payment such as disability, youth allowance, unemployment, parenting payment, sickness benefit - you name it.
Some situations require ID - we have a points system. A passport is worth the maximum number of points. A birth certificate or drivers licence comes in second - our licenses are photo IDs. After that comes Medicare card, bank statements, credit cards/debit cards, and last (ironically) is ID cards (other than drivers licenses).
With all this, we don't get many shops asking for phone numbers, drivers licenses etc just for a purchase. An exception to this is buying a mobile phone on a pay per month plan. difficult child 1 didn't have a drivers licence when he bought his, so his mobile phone is in my name.
But yes, various shops keep asking for our postcode (equivalent to Zip code). It is optional, we can refuse. I usually do because for us, they will get no useful information from our postcode. While most postcodes cover a suburb or less, our small twin villages here have to share a postcode with the nearest town 'as the crow flies" which in our case is the other side of the bay. That's an hour's drive away. The nearest town by road has a very different postcode. So any marketing they might plan, based on the postcode we give them, will miss its mark.
If you want to experience Big Brother in action, move to Australia.
Marg
I can't remember those many numbers.
and get some 
