I'm not sure who you mean when you say 'they had a fire all winter long'... wood ashes do contain small amounts of potash, but the actual potash content varies hugely on what type of wood/plant material is burnt to produce it. It also does not keep - if it becomes damp, any potash content leaches out quickly. The term 'pot-ash' originally applied to the substance left behind after soaking ashes from burnt plants in a pot with water - after evaporation, any white coating round the pot was potash and used as such.
The sort of potash most commonly used by gardeners is Sulphate of Potash, which is created chemically, and its used primarily around fruiting plants. Mined potash has chloride in it, and this is used much more in agriculture, often for cereal crops.
More info here http://acceleratingscience.com/mining/potash-a-look-at-the-worlds-most-popular-fertilizer/