A medial post is a higher density material within the midsole of a running shoe and is given the name of a dual density midsole. It was first introduced in 1984 in the Tiger X-Caliber GT running shoe.
The original purpose of the medial post was as one of a number of motion control design features in running shoes to control ‘overpronation‘ of the foot in the runner. The evidence of the medial posting being to able to do that is not very good, most likely because there are multiple causes of ‘overpronation’ and it is unlikely to be able to control all of those causes. For example, a tight calf muscle is a common cause of ‘overpronation’, but none of the motion control features in running shoes are going to be able to control the ‘overpronation’ from that cause as they can not address the tight calf muscles. The medial posting will have differing effects on different runners, depending on what is causing the ‘overpronation’ and assuming that they are actually overpronating and assuming that something actually needs to be done about it.
Where medial posting is particularly useful is that it will move the center of ground reaction forces medially. This will increase the lever arm that ground reaction forces will have to the subtalar joint axis. Those with a medially located subtalar joint axis would benefit from that.
Medial posting as a design feature in running shoes is being used less now, which is a shame as it does have practical uses in those runners that do benefit from it. It is of no use to those runners who do not need it.