How do we know that everybody's DNA fingerprint is unique?
First, remember that identical twins actually have the same genotype. So its not exactly true that everyone has a unique genome.

But to get at the heart of it, you're asking how I can be sure that I have a different genome than you, or even than say my brother. And moreover you're asking how these differences are obvious enough that they can be detected without whole genome sequencing.

The answer is that there is no mechanism to ensure differences, but that the differences arise from the enormous variability of our genome. Look at the major histocompatibility complex, or MHC, a cluster of genes that affect our immune system.

Just to simplify things, lets focus in on four genes in this cluster: HLA-A, HLA-B, HLA-C and HLA-DRB1. According to this site there are 2579, 3285, 2133 and 1411 alleles for each, respectively. If we make a naive assumption that each of us received a random assortment of these genes, that makes 2579 x 3285 x 2133 x 1411 = 25,549,791,000,000 possible combinations. The chance that we both have the same combination is then about 1 in 25 trillion, just considering 4 genes.

But wait, you say, surely some of these genes are not equally likely and surely they segregate together, making this chance much smaller. You would be right. But this consideration stops mattering very much when you account for the rest of the variable regions. They combine multiplicatively, so considering a fifth gene with only 2 possible alleles brings the possible combinations to 50 trillion.

If you consider just a few more genes, the numbers get unthinkably high. This pretty much ensures that, even considering a small sequence, we can be sure that the genetic fingerprint is unique.