it very well can, and it wouldn't be the first acid thing that does! (ACV, OJ etc.). What makes a substance alkalising is not much its ph but its content in minerals.
I just happened to read something that can give the explanation to this: urine can be considered structured water and therefore the minerals it contains are highly soluble. //www.curezone.org/forums/m.asp?f=213&i=106#i
A part from the Science of it: I can assure you, and you can try yourself if you feel like it, if your urine is sour, drink it (not too easy, I know), the next one will be for sure less sour. Besides, just after drinking the urine you will feel immediately better. The best urine to drink is according to everyone the first in the morning, which is also very likely to be the most acid of all day, but it is also the richest in minerals, and that's what matters.