What is causing/caused these brown circular stains in my cabinetry?
I'm pretty sure this is bleed-through of knots from underneath the paint. They have the shape of knots, and it looks like what you get when that happens.

If you want to be dead positive (or don't believe me) the only real solution is a scraper. Yes, its a bit on the nuclear side, but it will prove the existence of a knot underneath. Scrape off the paint in that area of wood. Pick an obscure spot to do so. Bring it down to bare wood, and you should see a knot underneath. Since the answer is to use a product like Kilz anyway to intercept the stain from bleeding through again, having to repaint that small area is no real issue.

Why can't you just paint over it? Well, you can, and for a time it will look like it worked, but the stain will come through regular paint. So you need something that a stain can't bleed through. In the old days, one might have used shellac as a stain killer, in fact, I think some stain killer products were based on shellac.