The same assumption (that similarity in DNA is evidence of common ancestry) is used in every Family Court in America.
The parts are basically the same, yet, we cannot create a human-chimp offspring. (Who has tried a chimp-human one?)
There are occasional claims of such things (Google "humanzee"), but no one has demonstrated this under laboratory conditions. (There are laws against such things.) Whether or not it would be possible, that doesn't mean that we and chimps don't share common ancestry. Tigers can only rarely produce fertile offspring with lions, and can never produce such offspring with leopards or cougars, but I don't know anyone who doesn't accept the fact that all big cats share a common ancestry.
Where is our CLOSEST animal critter?
DNA evidence shows it is the chimp. In fact, chimps are genetically closer to humans than they are to gorillas.
why can we not BREED with them?
See above. It's called "speciation."
And it means: Different Species
All the hollering that is why we can't breed, does not make it so.
4% (or 2%, depending on which 'Scientist' you listen to) DNA between critters and they can't breed?