It not only does, but it does in the same way we use to determine parentage in courts.
It's hilarious to watch creationists, who have absolutely no idea what the data is, act so confident in their declarations about the data. The above from you is a good example.
Transposons are a type of genetic parasite that replicates only in the genetic material of their host. However, unlike viruses, they don’t have genes for viral coat proteins and can’t cross cellular boundaries. Also, transposons come in two general categories: retrotransposons and DNA transposons. Retrotransposons replicate via “copy-n-paste” (they use RNA to make a copy of themselves, which is inserted elsewhere in the genome). DNA transposons move about via “cut-n-paste” (they use an enzyme to cut themselves out of the genome and then reinsert themselves somewhere else in the genome). In both cases the location of the insertion/reinsertion is random. This has been directly observed to have happened to many organisms (e.g. yeast, humans, bacterial, flies).
If the genetic material of the transposon is inserted directly into the host’s genome in a germ line cell (an egg or sperm), all the descendants of the host will inherit this material. Additionally, because the insertion is random, the only way two organisms would share the same transposons in the exact same locations is if they shared a common ancestor.
A common class of retrotransposon are SINEs (short interspersed elements). One important SINE is the Alu element. Alu elements are around 300 base pairs long, and are commonly used in paternity testing and in criminal forensics to identify individuals and establish relatedness. They are reliable identifiers because of what I discussed above, namely that the only reason two individuals would share the exact same particular Alu sequence in the same location is if they both inherited it from a common ancestor.
About 2,000 Alu insertions are specific to humans, and an even larger number are shared with other primates. But more specifically, in the human alpha-globin cluster there are seven Alu elements, and each one is shared with chimpanzees in the exact same seven locations!
Evolution of Alu family repeats since the divergence of human and chimpanzee. - PubMed - NCBI
So the same methodology that allows us to determine paternity and relatedness in courts of law also allows us to show that humans, chimpanzees, and other primates share a common ancestry. Thus, this bit of evidence “proves” human/primate shared ancestry in the same way the same evidence “proves” paternity in courts of law.
