R-L165 is a subclade of R-L51 so you two definitely have a common straight line male ancestor. But the real question is how many generations back is your most recent common ancestor (MRCA). The only thing you know from the 23andme test data is that you both have R-L51 in common. That means you have an MRCA on the straight male line sometime within about the last 6100 years. That does not really narrow down the relationship possibilitites. You could be brothers with your match or third cousins or your could be 200th cousins. Having R-L51 only tells you that it is somewhere in that range.
Now say you had taken the test with the V5 chip and it came out that you both had R-L165 in common. That would still only narrow the time frame down to within the last 4700 years which is not much of an improvement genealogically speaking.
To bring that time frame into a genealogical useful one you would both have to do a more refined Y-DNA test such as you might get in a Whole Genome Sequencing or one of the Y specific sequencing tests. That would give you a much more refined idea of what your two haplogroups are and would tell you much more definitively how far back in time you might have to look for your MRCA.
There is also the slim possibility that 23andme's analysis is outdated. The understanding of the Y-Tree is changing pretty rapidly as more Y-Sequencing tests are done. There could be a Y-SNP in the 23andme set that refines your haplogroup better than R-L51. On the isogg page you linked to in the question there is a link to a spreadsheet listing of all the Y-SNPs on the different 23andme chips. You could download your and your matches raw data from 23andme and compare them to make sure you have most of the V4 SNPs in common and that there is not one of those SNPs that indicates a more recent haplogroup than R-L151.