That does indeed look like an old well. The metal ‘pole’ is very likely to be the old draw pipe.
What you want to do with it is up to you. It’s yours. There are no rules or regulations that state you have to report it to anyone. I bang on people’s doors fairly often looking for olds wells, as lovewookie says, it allows a risk assessment of vulnerable receptors to be made if there has been a pollution incident nearby.
A shaft well like that is likely to be fed from the strata that it’s sunk into rather than rainwater infiltration, but it does rather depend on your locality. Without knowing your precise location I can’t comment on the geology, but it’s easy enough to find out about that stuff online.
I’d be careful about drawing conclusions of water table levels from well water levels too. There’s not always a correlation, especially in interbedded sedimentary strata.