What is causing/caused these brown circular stains in my cabinetry?

Recently moved into a 1950s-built home and have unearthed a number of brown, circular stains on the wood inside various closets. These rings don't appear on the plaster walls, but rather on the wood shelves or siding in the closet. Here is a picture of one such brown stain that appears on the wood along the top of the hallway closet.

I've seen these stains in a number of non-adjoining closets throughout the house. The brown ring is not wet or damp to the touch, nor does it smell funny. If I take a scraping tool and scrape off the paint the same pattern appears on the wood.

What's interesting is that the stain appears to permeate through the piece of wood. For instance, here's a picture of the top of a shelf in one closet - note the two brown rings:

And here's a picture taken from the underside of the same board - note the matching rings.

My first guess was that this was a water stain of some kind, but I am more doubtful seeing as it's always a ring (and not a full circle) and that it permeates cleanly from top to bottom of the wood. Is this a knot in the wood and has sap leaked out, perhaps? Is this mold? Something else? Any ideas?

Also, what do I do about these? Should I just paint over them? My concern is that it would be just dealing with the symptom of a problem; also, who's to say that the stain won't go through another layer of paint? Or do I need to rip out the shelves and side paneling and put new ones in? (The wood/interiors of the closets appear very cheap so it's not like I'm ripping out high quality, custom-built cabinetry.)
#Wood #Painting
Wood Painting