Although someone else said that the chances of getting prostate cancer a second time if you were 'young' were high, two different oncologists and a surgeon I spoke to said that detection of a brand new cancer after the initial treatment was very very rare.
Instead, the normal cause of any future biochemical failure was almost always because the initial treatment had failed. That could be because of distant mets, radio resistance, positive margins or something else.
So, whilst I agree that it's totally reasonable to conclude that if you still have a prostate, you can get prostate cancer a 2nd time, my takeaway is that recurrence (by definition) is much more likely as the result of the initial treatment failing rather than because an independently new disease developed.