If this is true, why can't we fuck animals today?
As several people have told you, it is physically possible for a male human to copulate with a female mammal of a few other species, but we are not sufficiently closely related to any of them for the act to result in fertilization of an ovum and pregnancy. The sperm and ovum will simply not combine.
But another equally serious impediment to this is that most female mammals are physically incapable of copulating when they are not in estrus ("in heat"). Their organs simply will not respond. Human females are almost unique in their ability to copulate when they are not fertile, and even when they are pregnant or nursing. Very few other mammals can do that. Chimpanzees and dolphins are the only ones I can name.
But if you meant to ask a different question, "Why can't we copulate with another animal and have offspring?" the reason is explained below.
By the way,
humans are animals, so when we have sex with another human, we are having sex with an animal. There are only six kinds of living things on this planet: animals, plants, fungi, algae, bacteria and archaea. We are clearly not members of the last five groups, so we must be animals.
How does the DNA know the difference, since virus can insert DNA?
The sperm and egg cells are simply the wrong shape to combine. They don't have connectors in the right places to link up.
In general, two organisms of different species can only crossbreed and have viable offspring if they belong to the same genus. This is why horses and asses, lions and tigers, wolves and coyotes, scarlet macaws and blue-and-gold macaws, etc. are capable of crossbreeding and producing hybrid offspring.
But
Homo sapiens is the only species within genus
Homo, so there simply are no other living animals with which we might be able to hybridize. The Neanderthals,
Homo neanderthalensis, were members of our genus and indeed crossbreeding was apparently common, since most of the population of Europe has a bit of obvious Neanderthal DNA, and people on other continents don't. But they disappeared, either from climate change or simply from being absorbed by our species. Today we have no relatives left.
That doesn't even make sense, as we did not evolve from chimpanzees.
Not exactly. But we and the chimpanzees come from a common ancestor. However, we have been evolving for so long (it's been at least 7,000,000 years since the two lines separated and our oldest known ancestor
Ardipithecus came into existence) that we are no longer members of the same genus. Both the Bonobo chimpanzee,
Pan paniscus, and the "true" chimpanzee,
Pan troglodytes, are members of the genus
Pan, and (as far as I know) it's possible for them to hybridize. But they cannot hybridize with humans; our DNA is too different.