It looks like quite a new fence, perhaps it was put up by builders. In which case they have probably lobbed a huge barrowload of readymix concrete into the ground.
This is very tiresome as it is a lot of work to take it out and put a new post in.
I strongly recommend concrete posts and gravel boards, as they do not rot or warp. It is possible to use masonry paint to make them blend in with your fence stain and look very nice.
Your post appears to be next to the house. If it is very difficult to dig out the old one, or even to add a concrete repair spur, can you bolt a new post to the house wall? This is quite normal. You will need very long steel bolts, or , if you can let the fixing show in the inside surface, galvanised studding going right through the wall, with a nut on each end. This is very strong and does not come loose. You might have to pack it out a bit to meet the existing fence. Do not use bolts or screws close the the edge of a wall as the bricks may split or crack
Depending where the damage is to your old post, you might even be able to keep the upper part of it, and bolt it to the wall using packers.
If you are taking the old post out, you may be able to prop up the fence while you are working, but wind may take it over and cause more damage.
