Hardwood ash, or any ashes, for that matter is/are a fertilizer in that
it/they are a source of potassium that can be readily used by plants. The
problem with ashes used as a fertilizer is that it contains potassium
hydroxide which is only a slight step less toxic than lye (sodium
hydroxide). Ashes also contain a bit of sodium hydroxide plus some
insoluble phosphorus. The phosphorus can be used by plants only after soil,
bacteria convert it into a more soluble form. Ashes contain nearly zero
nitrogen.
Really, though, ashes is/are close to the worst form of fertilizer
unless your soil is too acid and you want to sweeten it.