You and me are different at DNA level. My eye color gene is different from yours. So, my DNA is different from yours. How can a scientist identify a certain gene in a chromosome (and its function) if chromosomes are different? How can we talk about "the" human DNA when nobody share the same (except twins)? Please, this is a practical question about how scientists identify genes, not a philosophic one (a question that bother me every time I read about discoveries in genetic)
First of all: We are not very different on the genetic level - the identity is somewhere around 99.6 to 99.9%. See here for details. If this wouldn't be like this, things like blood transfusions or organ transplants wouldn't work.
To identify genes there are different routes. "In the old days" (meaning before the possibility of massive high throughput sequencing or DNA microarrays), genes were usually discovered when they were connected to a disease which had a noticable phenotype. Researchers then tried to find out which protein or pathway was affected and from there went backwards to identify the genetic region. This way of discovery was relatively slow. An example for this kind of identification is the identification of the Mitf gene which is important for the development of pigmentation.
If you sequence DNA (doesn't matter here if these are complete genomes or only parts of it) you can identify genes based on homology with already known genes. You can also predict the presence of genes in a certain sequence based on regulatory sequences in the region before the gene (the promoter). These sequences are known and also highly conserved, so this gives a good estimate. Predicted genes usually need to be verified experimentally.
What is done today are the so called "genome wide association studies". Here you take a big cohort of people which all share one phenotype (for example blue eye color). Then you take a second group of people which do not show this phenotype and analyse their genomes (usually by sequencing or SNP genotyping). Then you compare the two groups to find the differences. Ideally you can then more or less directly identify a causative mutation which is responsible for the phenotypic difference between the two study groups. This usually needs also to be verified further.