Are all Y-chromosomes the same?
Actually, no. There are also recombination prone regions of the Y chromosome that recombine and exchange material with X chromosomes, and these are called pseudoautosomal regions (PARs). 

Y chromosomes can be used similarly to mitochondrial DNA to build up profiles of ancestry, but the sequences used for this purpose lie outside PARs, in the non-recombining region.

What about seqeuences outside PARs, do they show genetic variation?

As pointed out in the comments, PARs comprise only 5% of the Y chromosome, but even in the non-recombining regions of the Y chromosome studies in well-characterised populations have found that the mutation rate can approximate autosomal mutation rates in some regions of the chromosome, and exceed them in others. 

What processes, other than recombination, drive mutations?

There are loads of different mechanisms by which mutations in general can be generated, including exposure to UV, free radicals, tobacco smoke, aristolochic acid, and perhaps most importantly spontaenous deamination of methylcytosines, which is an age related mutational process, proofreading errors inherent to DNA polymerases. Also, it is well known that in sperm, with age the mutation rate goes up throughout the genome. While UV is unlikely to be a prominent mutagen as was pointed out to me in the comments insofar sperm are concerned, factors like smoking still play a role. 

An illuminating read about mutational processes and the mutational footprints they leave in genomes, as understood from studying cancer genomes, is here.