Is he healthy and happy most of the time? If his colouration does not change and he will still respond if stimulate by something like food i would say he is probably just resting, quite a few betta's do this.
my beta used to lay on the internal filter, resting and letting the filter do the work. also hid behind slate and deep under bogwood with my clown various loaches
He looks cozy
I haven't had a betta for a long time, just a few months. I've never seen him lay down on his side so far. I've seen him using his fins to perch like a bird on a plant or decoration but that's about it. But I guess they can be different. If his fins, colour, behaviour and appetite are the same then it could be that he's found out the most comfy spot in the tank.