"Koalo" (Koala - Phascolarctos cinereus)
Sure, it lives its life in trees, dines almost exclusively on a plant genus that is incredibly non-nutritive and toxic to most animals (Eucalyptus), and the males have a two-pronged penis, but the koala has more in common with humans than you might think.
For one, they have lots of problems with venereal diseases, including one that’s so closely related to the human strain it can be transmitted across species - chlamydia. However, in koalas, chlamydia is present even in most healthy animals, and it’s only when the animal gets stressed or otherwise weakened that it manifests as disease. There’s currently a huge uptick in the numbers of koalas infected with chlamydia, causing mass sterility and, in many cases, death.
On a less dire note, koalas are the most distantly-related mammal to display "dermatoglyphes" - fingerprints with ridged loops and whorls, like humans have. In fact, it’s almost impossible to tell the difference between the two under a microscope. As Homo sapiens and Phascolarctos cinereus diverged over 70 million years ago, it’s clear that this is a case of convergent evolution, developed to help the koala grip onto branches and tree trunks.
Aracana, or, The museum of Natural History. George Perry, 1811.