Is the prostate problem unique to humans?
Dogs get benign prostatic hypertrophy, so, yes, other animals (in fact most old male dogs, so that would mean a whole lot of other animals) do get BPH.


  The most common conditions affecting the canine prostate include benign prostatic hyperplasia, prostatitis, prostatic cysts, and prostatic neoplasia.


However, you're almost correct.


  [T]here is no risk for prostate cancer in any other aging mammal except the dog.


Though this article addresses cancer, the same appears to be true for BPH.

The prostate and breast appeared at the same time with the origin of mammals (possibly as long as 195 million years ago). While all male mammals have a prostate, seminal vesicles only appear in  primarily herbivorous species with the exception of man. The human male has seminal vesicles and consumes meat. Thus, it has been postulated to be a dietary cause. Bonobos, on the other hand, exist primarily on a high fruit and vegetable diet. They don't get BPH.

It isn't known what dogs ate before domestication (were there 'dogs' before domestication?) But it's postulated that ~10-15,000 years ago, humans (and their newly domesticated partners, dogs) dramatically altered their diets, breeding animals for food.  

Why haven't we evolved a solution? Well, on the one hand, if BPH is only 15K years old, that's only a fraction of a blink in the human evolutionary scale of things, so not enough time to "evolve" a way around it (maybe reroute the urethra?)

On the other hand, we have. We've evolved medications and surgery for treatment of prostatic hypertrophy.

Similarities of prostate and breast cancer: Evolution, diet, and estrogens
New concepts in tissue specificity for prostate cancer and benign prostatic hyperplasia