The answer is complicated. In the early days of molecular phylogenetics, it was extremely difficult to demonstrate whether chimps or gorillas are our closest relatives. This is because the common ancestor of chimps and humans lived between 5 and 8 million years ago, and the common ancestor of gorillas and humans (and chimps) lived around the same time, just a little older. This means that there was a rapid split of three lineages, and it was difficult to disentangle them. As molecular phylogenetics advanced, we could generate larger datasets, and it because clear that chimps and humans were sister taxa. The smoking gun that demonstrated that chimps and gorillas were not one another's closest relatives was found when sequencing across the chromosome region in humans that corresponds to where two of our ape ancestors chromosomes merged. Orangutans, Chimps and Gorillas have 24 pairs of chromosomes, humans have 23. Sequencing across this region shows the exact splice site, which can be associated with the same regions on two chimp or gorilla chromosomes.
To answer your specific question, if you sequence any two genes between chimps and humans, you will find that the sequences are almost identical, with perhaps 1 in 100 nucleotides being different. If you sequenced a human and a gibbon for the same gene, you would find more differences. But finding that the sequences are similar is not the same as finding relationship, which requires sequencing multiple genes from multiple animals. For example, some genes between humans and rabbits would be extremely similar. So inference of relationships takes a lot of data from a lot of different animals. As for paternity testing, these practical tests are based on a different technology, targeting many DNA repeat regions or single nucleotide polymorphisms at specific sites that vary among human populations. But since the test is prescribed, and established, they are cheap, quick, and easy.