The Chromosome Browser image you've displayed does show half matches, however, you should think of this display as a list of half matches, not a display with one half of a chromosome pair on top and the other half on the bottom. For instance, if you were comparing against five other people, and you and those five people all shared an identical segment, then there would be a stack of five colored blocks. Obviously, such a display would not be saying you have 5 halves on that chromosome pair. I can't quickly come up with an example showing 5 overlapping matches, but here is one with 4.
In this image, we see that the person whose test we're viewing matches 4 people over a common region of this chromosome. What we don't know is whether the all five people share the same segment on one half of their chromosome pair, or whether some of them match on one half of the test subject's chromosome pair, and some of them match on the other. To answer that question requires different tools that help you triangulate, and determine which people match in common (like with the "in common with" filter on the FamilyTreeDNA FamilyFinder Matches page).
As further evidence that the Chromosome Browser is not attempting to display the two halves of each chromosome pair, but is in fact simply displaying a list of matches, examine the display of a comparison to a single person:
Here, it is showing that in a comparison to a single person, there is a half match with that person in the area of the chromosome marked by the orange block. It isn't saying that there is a full match (both halves of the chromosome pair) with that person.
So, in the image you've shown, if that were the X chromosome for a man, it would indicate that the person and the two matches all have matching DNA on the X chromosome where the orange and blue segments overlap.
In reality, it's a little more complicated than this, because testing services such as FamilyTreeDNA, 23andMe, and AncestryDNA do not produce phased data, but for the purposes of your question, we can wave off this detail.
The X chromosome is not displayed any differently than the other chromosomes - if a man matches anybody on the X (male or female) the matching segments are displayed. If a woman matches anybody (male or female) the matching segments are displayed. The only difference is in the matching algorithm. For men, there is only one allele value for each SNP to use for matching. Women have two allele values for each SNP (similar to the paired values of the other chromosomes).
For autosomal (non-sex) chromosomes, we have 22 chromosome pairs (44 total chromosomes). Since they are pairs, we have two allele values for each SNP (a specific location on the chromosome). For women, the X chromosome matching works just like the matching for any other chromosome pair, because women have a pair of X chromosomes. In men, it's a little special, because they don't have a pair of X chromosomes - they have a single X chromosome and a single Y chromosome. So, for the X chromosome, men have only a single allele value for each SNP. This can be seen in the karyogram image on this website. There is a minor difference in the matching algorithm for men, because there is only one allele value to consider when checking each SNP for a match with the other person.