Is DNA so unique that 50% of it is enough to identify a single person?
The comment from @User137 pretty much gets it: the mix of all sperm cells in even a small sample will contain the whole genome on average. So in the hair example in the comments above, there are probably a good number of dead cells hanging onto the hair, enough for analysis. Meiosis will redistribute the DNA sequences inherited from mom and dad randomly, so each sperm is in theory a unique patchwork of that person's genome, but overall the genome will be covered well. Single-cell work, on the other hand, is technically very challenging.
[edit: here is an article describing methods for low-input analysis of forensic samples. Unfortunately paywalled.]
To directly answer the OP on uniqueness: absolutely a single sperm is that unique for each person, in principle. There are lots and lots of genetic variants in each human, and everyone represents a unique mix. It is estimated that 1/1000 places in the genome of each person differ from any other person, on average (for substitutions). there are 3*10^9 places in the genome for this to be the case, which gives you a very large number of differences, ASSUMING that you can actually ascertain all those differences. [edit: for a discussion of these stats see here.]
The technical problem with forensics is that often they only look at a few places in the genome (microsatellites usually), because it is relatively expensive and hard even with recent technology to look at more than that in any detail. This leads to the non-trivial possibility of misidentification with the standard battery of loci. This possibility never goes away entirely, but it becomes very small with more data. I am not a forensic scientist, possibly they have gotten more comprehensive or the methods have changed in recent times.
[edit: here is a document that has what appears to be a readable introduction to DNA forensics, including technical discussions of some drawbacks of marker panels used by for instance the FBI of the USA.]
It would be more difficult to identify someone from a single sperm cell, rather than other cells contained in their semen.
However, if we are just comparing the DNA from any diploid cell in the body, you run electrophoresis to compare DNA samples.
For example, if you wanted to establish the paternity of a child. You would collect a sample of the child's DNA, the mother's DNA and the possible fathers' DNA (usually all from cheek or hair cells).
PCR would be carried out to replicate each sample into a large quantity. You would then use the same restriction enzyme in each sample to cut the DNA in specific places along the sample.
Each sample is placed in an individual well at one end of agarose gel, immersed in a buffer solution, and a current passed through (cathode at the well-end, anode at the opposite end).
As DNA is negatively charged due to phosphate groups, it travels towards the anode, through the gel. Shorter fragments of DNA travel faster in the set time the electrophoresis is run for.
Southern blotting is then used to establish a banding pattern of the fragments of DNA - you rule out the bands that the child and the mother share. To establish paternity, you identify bands that the child shares with each possible father. The father whom the child shares the most bands is the most likely father.