Cross section through the right arm question
Based on the location of that cross section in the upper arm the vein in question appears to be the basilic vein. The basilic and cephalic veins are the largest veins (most of the time) in the upper arm. Here is a nice diagram from wikipedia that shows both. The cephalic vein is much more superficial than the basilic vein, as you can somewhat appreciate in the diagram below.



I think the label "brachial vein" is probably an honest case of mislabeling. 

This is another figure that supports that vein in the image in the question as the basilic vein as well:


From this image (Gray's Anatomy), you can appreciate that the basilic vein is the large and deeper vein that runs near the brachial artery. In the case of many arteries and veins there are similar names, but that does not appear to be the case in this instance. 



Clinical Correlate: In vascular and/or transplant surgery, when making an arteriovenous fistula for dialysis access (for kidney failure), there are a number of common routes of access. Two of the most common are the (1) brachiocephalic fistula, which is formed by surgically connecting the brachial artery to the cephalic vein, and (2) brachiobasilic fistula, which is created by connecting the brachial artery to the basilic vein. Some additional information and a picture can be found in this open access article from Maya et al entitled Outcomes of Brachiocephalic Fistulas, Transposed Brachiobasilic Fistulas, and Upper Arm Grafts .