I had a fence. It was destroyed in a hurricane. I thought I just would need to put new posts in an existing 2 feet hole. But the holes there were not 2 feet. So I started digging holes around where the posts were. And I noticed cement structures going deep into the ground. Could this really be quick set concrete? It is stronger than a rock. I can not get these structures out of ground (they go real deep). Right now I am just digging holes around the concrete, which will make the fence uneven.
It looks like a standard 4×4 post with concrete poured around it. The square in the middle should have some wood left to be sure. The hole may go 36-48" down depending on local code even deeper. I usually use a tractor and a chain to pop old fence post out when on a job but on my place I just shifted the fence post several feet and left them in the ground.
I agree with Ed Beal: this appears to be (have been) just a regular wood post set in concrete. The common method of pouring dry mix into a hole with water poured over the top can work, but if the dry mix is well-blended with the right amount of water and then poured into the hole the resulting concrete will often turn out much stronger. That may be what you're up against.
Going beyond the "what is it?" question that was asked.. You'll get those out with the least effort by lifting them vertically. A small tractor like a skid loader or mini excavator will do nicely. Put a few wraps of chain around the concrete a few inches below the top and pull upward. (by "a few wraps" what I really mean is to tie a clove hitch with the chain.)
When heavy equipment isn't an option I've used a 2-ton chain hoist. I lash a tripod with 2x4 boards 8 or 10 feet long and suspend the chain hoist from this directly above the concrete. The whole chunk usually lifts out readily, and once it's up on the surface of the ground it can more easily be broken down if necessary.
If the remaining wood in the center is not totally rotten, you can often drive a lag bolt (huge wood screw) through the gap in one link of a heavy chain (tow chain), then pull up on the chain with a tractor or farm jack to pull the post out.
Or if the concrete is tapered so it's smaller lower down, just wrap the chain around it once or twice and hook it back to itself. It's amazing what you can pull out like that.
100% agree that it's a 4"x4" post setting, I just set one myself this last weekend! If it helps to know I dug my hole a little over 2 feet deep and then dumped three 50 lbs bags of quick curing Quikrete in there. So mine is anchored by 150 lbs of concrete.
In your case, considering that it's just a fence post I'd be surprised if that setting goes deeper than a foot and a half or two feet, but you can probably expect that plug of concrete to be about 150+ lbs.
I'm confident you can extract it in pieces if you drill some holes in it and take to it with a sledge and a demo chisel. Alternatively you might want to research renting a jack hammer in your area. Whatever you do...don't forget eye protection and ear plugs.
@GregHill's suggestion to use an excavator or skid loader is further than I would go, however I wonder if his technique would work if you swapped in a 2 or 3 ton shop jack in place of the machinery? You'd need some wood under the jack to keep it from sinking into the dirt. And I'd also use a couple cinder blocks to get additional clearance. But it might be enough to lift the plug just enough to wack it in the side with a sledge, then remove the individual pieces!
If you have the right tools (hammer drill,long chisel ) you can remove the rotten wood ,using the same pocket to set your new 4x4 post,I've done this and it takes a little time but it works and you can maintain your same spacing and maybe reuse some fence sections