I don't know of any DIY truly hidden fastener solutions for this application but there are many proprietary systems for doing so. Just search for "invisible mounting wood slats".
For decks and other larger dimensioned projects you can use a jig to come in at an angle on the inside edge, but I don't see that working out well for this application. You might be able to adapt one of the decking systems out there, but few of those are truly "hidden" as they often require special fasteners that are tucked in between the decking, or even special decking. This is clearly not what you are looking for with this kind of cleat style project.
So, I think you are going to have to go with one of the fancy proprietary hidden fastener systems sold for these sorts of commercial applications.
Though, the photo in your question is using staples or brads of some kind, which is a pretty clean install.
I should mention that I have owned a converted van in my life, and relying on friction (such as a keyhole) or glue is a Bad Idea. Vans flex. A lot. The stuff in vans gets thrown around, even on ordinary roads you can expect to want to take vans. A lot. Those joints will have to allow for this. And you cannot expect gravity to always work in your favour. A keyhole, when done right, is sort of a good friction fit. But it can easily be overcome.
And you should consider how to make it field repairable. Some fancy expensive method that requires special fasteners or tools is a bad idea. You will be repairing this on the road, eventually. Being able to do that with only a handful of screws and a driver of some sort is worth much more than the interior being Instagram ready.
Since the application here is clearly a storage solution, you have to treat it like one. If you make something that looks like French cleats, then you have to build it like French cleats. Assume that this van will be subject to forces twice what you feel on an emergency stop. Will it release several kilos of wood and tackle to the back of your head if that happens?
You 100% need a positive mechanical fastener than can handle at least a few Gs fully loaded. This is a place where form must follow function. Your constraint must be safety and durability, not cosmetics. And then see what you can design.
A rule of thumb is to treat this build like you are building a boat. Things need to be lashed down, and the things you are lashing to have to be connected to something else with a lot of fasteners.