This is "sticker stain". Sticker stain is fairly common and often unavoidable, although the nature and severity of it varies naturally.
Is there any treatment or method to remove or reduce the discoloration circled below?
Soon as you joint/plane or heavily sand the surface (which of course is the norm for wood rough from the mill whether it has been stored stickered or not) those will go, along with most or all of the rest of the 'rustic charm' the boards have.
Or do I just have to live with it as “part of the charm”?
If the plan had been to use the wood as-is then yes, you may have to live with them. Especially as some are light areas, where the wood was actually protected by those stickers and not actually stained by the sticker itself, or water wicking underneath them.
You could try pressure-washing the wood, or treating it with some kind of deck cleaner. There's no guarantee of success in either case, but I'd imagine you'd see a marked improvement with either approach. See previous Answer showing the effectiveness of pressure washing exterior wood, Can I run pressure treated deck boards through thickness planer?
No help now but for possible future use, this is why some people use stickers with a triangular section. Obviously you still have a flat which could lead to exactly these marks, but at least one side of each board will have a very minimal contact area so little or no marking will be present.
See more on air drying and storing wood in stacks in this previous Answer, What wood species should I use for making stickers?