I have often stared at documents like gas bills and been amazed to see things like account numbers or other identification numbers as long as 18 or 20 digits. There are only about 7 × 109 people in the world. Some account numbers are so long you could give separate account numbers to every member of the population on a billion planets with populations like ours. Those numbers could record the addresses and ages and incomes of the customers instead of just being random digit strings. But we don't do that. The information society that people get so worried about — the world in which The Government knows all your details and tracks everything you do — hasn't arrived yet, and probably never will. We're not that organized as a species. We waste too much time and too many of our computational resources keeping track of pointless random digit strings and being unable to relate them to each other.
This is an interesting misconception on a couple of levels. First, most long strings of numbers aren't simply random. They encode information in various portions of the digits, and they usually (though not always!) do it for good reasons. One of the most common reasons is to make the numbers comprehensible to trained human beings. If the first two digits of a part number indicate that it's a TV set, for example, that can make life a lot easier for a clerk who's entering the order. People frequently write down part numbers incorrectly, and human-readable chunks that don't match the English-language description can alert order-entry clerks that there's a problem. This in turn saves a ton of time and trouble remediating bad product shipments. I can attest from personal experience that this ability to decipher chunks of long alphanumeric strings comes in handy in a million different ways.
(Another example: yesterday Marian and I bought a new refrigerator. It turns out that in the model numbers used at Sears, "3" means stainless steel and "2" means white. We were able to catch a mistake before it happened by knowing this and mentioning it to the sales person who was helping us.)
Second, the fact that these strings of digits are or aren't random, and may or may not be longer than they need to be, has exactly zero effect on the government's ability to track everything we do. It does not imply any lack of organization, nor does it make anyone's job harder. The computational resources used to keep track of strings of digits is probably something like 0.0001% of the world's computational resources, and believe me, the fact that IP addresses are long doesn't slow down the NSA one bit. However, the fact that, in this case, they are largely random, slows them down a lot. There are times when we should praise randomness, and this is one of them.