People have added in wood ash to their garden for many years. I always thought it was just to add potash to the soil, and I barely made it through college chemistry, so I don't understand the process the wood goes through to make it into liquid smoke/vinegar. But maybe we old folks have been adding wood ash for more benefit than we realized.
... has me wondering . "Acetic acid only contains carbon hydrogen and oxygen ..." Okay, but that's only 5% or so of vinegar. I will look at your links ...
as the results of making biochar? depending upon the water content this can be a fuel. otherwise, acidic enough to use as a surface amendment spray perhaps on mulch for blueberries. i would not add it to a more general use garden soil because my aims in garden soil is to encourage diversity not to destroy it.
wood ash is a whole different material, and used in moderation is pretty good for some gardens. i mix it with my organic materials and clay when i am redoing parts of gardens as i have organic materials to use up. we don't burn wood for heat here but we are lucky enough to have some friends in town who do burn wood and they save the ashes for us - otherwise they'd be throwing them away in the trash to go to the landfill.