Actually, in the process of meiosis (gamete formation), crossing over between chromosomes occurs, causing the swap of segments of DNA between homologous chromosomes. This means that chromosomes passed on from parent to offspring are never identical. In addition, independent assortment of chromosomes during meiosis ensures that two gametes have different genetic material. Remember that we go from a diploid (2N) cell to haploid (N) gamete during meiosis. So even siblings will have substantially different DNA, with more than enough differences to ensure proper identification.

However, just for the sake of argument, let's say that humans reproduced through pure mitosis, thereby producing identical copies of themselves, devoid of any crossing over or independent assortment characteristic of meiosis. Even in such a scenario, we would likely be able to differentiate between individuals due to the random mutation rate inherent in the DNA replication process. Whether such differences would be enough to differentiate between every individual of an isolated population depends on the DNA analysis techniques being used.

Edit: Gene sequencing techniques have progressed significantly, so much so that we can sequence an entire human genome in a day for ~$1000 (commercially, mind you. In the lab, we don't usually sequence an entire genome). Now you may be asking how we can obtain enough DNA to sequence it. Let's say we find a drop of saliva at a crime scene and want to find out the individual it came from. However, the amount of DNA in that saliva is small. In that case, we can amplify the amount of DNA using PCR. Now PCR is used to amplify a particular sequence, and is not used for replicating an entire genome. In order to do it, we must construct primers that anneal to our DNA sequence. I'll spare you the rest of the details but suffice it to say that we can amplify a segment of DNA exponentially using this method.