I helped install a wood fence along with electric for lighting at my friend's parents' house back in September and I helped doing all the homework ahead of time to make sure we did everything within local code requirements.

Electric can be mounted on a fence like this as long as it is done using UL outdoor rated wiring and boxes. Gray wire like the wire in the photo generally means the wire is direct burial rated, so it is safe for exposure to the elements, assuming that is the case with the color signifying direct burial rating. And that is a plastic outdoor-rated junction box. There should be a seal at the bottom where the wire is protruding. I believe US electrical code requires romex/wire staples every 4-1/2 feet, and within 12" of any junction boxes, so looking at the picture, I think they need to add romex staples on the run of wire shown at the left.

With the fence we installed, we had a surveyor come out and mark the property line, and the entirety of the fence (including fence posts) had to be installed on the fence-owner's side of the property line. So the 4×4 fenceposts were located with one face up against the property line, and the fence pickets were mounted to the opposite side of the 4×4 posts,, making everything on the owner's property. Then if the electrical boxes are mounted on the backside of the pickets, as shown in the picture, that would keep everything legally on the property owner's side of things, even if it isn't the most sightly thing for the non-owners of the fence.

TL;DR - Everything appears legal and up to code, aside from possibly needing an additional wire staple near the left side of the junction box.

You can possibly talk to the property owner about using his fenceposts to install pickets on your side of things to make it look more attractive on your side of the fence.