Well, that's explained in (just a small portion of) Ichneumon's Legendary Post #52:
Human/ape common ancestry:
This is just a taste of the massive amount of evidence for ape/human common ancestry, the amount for evolution in general (including between different specific animal families) would (and does) fill innumerable encyclopedias worth of volumes:
Background: Retroviruses reproduce by entering a cell of a host (like, say, a human), then embedding their own viral DNA into the cell's own DNA, which has the effect of adding a "recipe" for manufacturing more viruses to the cell's "instruction book". The cell then follows those instructions because it has no reason (or way) to "mistrust" the DNA instructions it contains. So the virus has converted the cell into a virus factory, and the new viruses leave the cell, and go find more cells to infect, etc.
However, every once in a while a virus's invasion plans don't function exactly as they should, and the virus's DNA (or portions of it) gets embedded into the cell's DNA in a "broken" manner. It's stuck into there, becoming part of the cell's DNA, but it's unable to produce new viruses. So there it remains, causing no harm. If this happens in a regular body cell, it just remains there for life as a "fossil" of the past infection and goes to the grave with the individual it's stuck in. All of us almost certainly contain countless such relics of the past viral infections we've fought off.
However... By chance this sometimes happens to a special cell in the body, a gametocyte cell that's one of the ones responsible for making sperm in males and egg cells in females, and if so subsequent sperm/eggs produced by that cell will contain copies of the "fossil" virus, since now it's just a portion of the entire DNA package of the cell. And once in a blue moon such a sperm or egg is lucky enough to be one of the few which participate in fertilization and are used to produce a child -- who will now inherit copies of the "fossilized" viral DNA in every cell of his/her body, since all are copied from the DNA of the original modified sperm/egg.
So now the "fossilized" viral DNA sequence will be passed on to *their* children, and their children's children, and so on. Through a process called neutral genetic drift, given enough time (it happens faster in smaller populations than large) the "fossil" viral DNA will either be flushed out of the population eventually, *or* by luck of the draw end up in every member of the population X generations down the road. It all depends on a roll of the genetic dice.
Due to the hurdles, "fossil" retroviral DNA strings (known by the technical name of "endogenous retroviruses") don't end up ubiquitous in a species very often, but it provably *does* happen. In fact, the Human DNA project has identified literally *thousands* of such fossilized "relics" of long-ago ancestral infections in the human DNA.
And several features of these DNA relics can be used to demonstrate common descent, including their *location*. The reason is that retroviruses aren't picky about where their DNA gets inserted into the host DNA. Even in an infection in a *single* individual, each infected cell has the retroviral DNA inserted into different locations than any other cell. Because the host DNA is so enormous (billions of basepairs in humans, for example), the odds of any retroviral insertion event matching the insertion location of any other insertion event are astronomically low. The only plausible mechanism by which two individuals could have retroviral DNA inserted into exactly the same location in their respective DNAs is if they inherited copies of that DNA from the same source -- a common ancestor.
Thus, shared endogenous retroviruses between, say, ape and man is almost irrefutable evidence that they descended from a common ancestor. *Unless* you want to suggest that they were created separately, and then a virus they were both susceptible to infected both a man and an ape in EXACTLY the same location in their DNAs (the odds of such a match by luck are literally on the order of 1,000,000,000,000,000,000 to 1...), *and* that the infections both happened in their gametocyte cells (combined odds on the order of 1,000,000 to 1) *and* that the one particular affected gametocyte is the one which produces the egg or sperm which is destined to produce an offspring (*HUGE* odds against), and *then* the resulting modified genome of the offspring becomes "fixed" in each respective population (1 out of population_size^squared)...
Then repeat that for *each* shared endogenous retrovirus (there are many) you'd like to claim was acquired independently and *not* from a shared ancestor...
Finally, you'd have to explain why, for say species A, B, and C, the pattern of shared same-location retroviruses is always *nested*, never *overlapped*. For example, all three will share some retroviruses, then A and B will both share several more, but if so then B *never* shares one with C that A doesn't also have (or at least remnants of).
In your "shared infection due to genetic similarities" suggestion, even leaving aside the near statistical impossibility of the infections leaving genetic "scars" in *exactly* the same locations in independent infections, one would expect to find cases of three species X, Y, and Z, where the degree of similarity was such that Y was "between" X and Z on some similarity scale, causing the same disease to befall X and Y but not Z, and another disease to affect Y and Z but not X. And yet, we don't find this in genetic markers. The markers are found in nested sequence, which is precisely what we would expect to see in cases of inheritance from common ancestry.
Here, for example, is an ancestry tree showing the pattern of shared same-location endogenous retroviruses of type HERV-K among primates:
This is just a partial list for illustration purposes -- there are many more.
Each labeled arrow on the chart shows an ERV shared in common by all the branches to the right, and *not* the branches that are "left-and-down". This is the pattern that common descent would make. And common descent is the *only* plausible explanation for it. Furthermore, similar findings tie together larger mammal groups into successively larger "superfamilies" of creatures all descended from a common ancestor.
Any presumption of independent acquisition is literally astronomically unlikely. And "God chose to put broken relics of viral infections that never actually happened into our DNA and line them up only in patterns that would provide incredibly strong evidence of common descent which hadn't actually happened" just strains credulity (not to mention would raise troubling questions about God's motives for such a misleading act).
Once again, the evidence for common descent -- as opposed to any other conceivable alternative explanation -- is clear and overwhelming.
Wait, want more? Endogenous retroviruses are just *one* type of genetic "tag" that makes perfect sense evolutionary and *no* sense under any other scenario. In addition to ERV's, there are also similar arguments for the patterns across species of Protein functional redundancies, DNA coding redundancies, shared Processed pseudogenes, shared Transposons (including *several* independent varieties, such as SINEs and LINEs), shared redundant pseudogenes, etc. etc.
...actually there's more, but that should do it.