In human males the urethra goes through the prostate rather than around it, which tends to create problems for older men as the prostate enlarges. Is this only a human problem or do other animals, primates in particular, have the same problem?
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.
Why prostate surrounds urethra? The human prostate has different zones that form separately in the embryo from the urogenital sinus, but later fuse. The mouse "prostate" is actually several separate glands that remain distinct. Although there has be a temptation to associate specific humane zones with specific mouse types of prostate, the current feling is that this is not fully supportable. Why the human zones have fused remains unclear. See: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4968575/
Regarding evolutionary pressures: In order for a trait to be selected against, it needs to be "active" or forceful during the period of reproductive activity. That's not the case for BPH. It occurs mainly after the time of typical procreation for humans in pair bonds. There are very few societies in which 60+ year-old men are a "reproductive" force. Furthermore BPH is not fatal, so even if we older men were actively engaged in siring more offspring, we could still be doing so into our 60's and 70's.