I assume it's tracking chromosomes. Humans have 46 chromosomes, and each chromosome is basically a copy of one from our parents, plus some mutations. You can count the average rate of mutation per generation (which is very low, something on the order of 100-200 total mutations in the entire genome), and then count the differences between one chromosome and another chromosome. When comparing genomes for two individuals, you'll get a low number of differences for some chromosomes and a higher number of differences for others.
So when I hear 25% similarity, I'm thinking that it means that 25% of the chromosomes for person A are very similar to chromosomes for person B, and that the remaining chromosomes are not as similar. This is just relative similarity, the overall similarity will be quite high.
You're right about the relative similarity (and that's the key point), wrong about the number of chromosomes being involved in the 25% calculation. It's about the proportion of bases (letters in the sequence) that can differ between two people that actually differ between these two people. See mbreese's comment: https://news.ycombinator.com/item?id=8293159
So when I hear 25% similarity, I'm thinking that it means that 25% of the chromosomes for person A are very similar to chromosomes for person B, and that the remaining chromosomes are not as similar. This is just relative similarity, the overall similarity will be quite high.