The phrase 'belt buckle to shoe sole' is an idiolectic, truncated version of the formulaic 'top to bottom'. That same formula also sponsors such phrases as 'head to foot' (or 'head to toe'), and etc.
Precursors of Biden's truncated version include, among those already mentioned, "collar to shoesole" (paywalled; The Mahoning Dispatch, Canfield, Ohio, 15 Mar 1912, p. 6) and, the more likely source of Biden's uncle's putative expression, "hat to shoe-sole" (paywalled; The Lyons Republican, Lyons, Kansas, 13 Sep 1888, p. 1), which appeared in the popular press response to an 1888 speech by Ohio supreme court justice, senator, and vice presidential candidate Allen G. Thurman.
In Thurman's speech, the formula was considerably embellished:
These advocates of protectionism are all at once seized with
A WONDERFUL SOLICITUDE
for the laboring man, and they want a high protective tariff, not to benefit the manufacturer, according to their statements, but to benefit the working man. He is the man they seek to protect. I would like to know how taxing a laboring man on everything from the crown of his head to the sole of his feet is going to enrich him. And yet this is exactly what this tariff tax does, and they say that is for the benefit of laboring men. Because the capitalist employer will make more money and can pay higher wages? Did you ever know him to do it?
(Paywalled; quoted from a report in The Salt Lake Herald, Salt Lake City, Utah, 23 Aug 1888, p. 1.
Biden's truncated version of the formula has an air of insider, or family, humor. I suspect that the pertinence of his sense to only half of the whole, the bottom half, is owed to Biden's personal and family history, wherein his more recent modest wealth is founded on middle-class beginnings and a political affinity for working-class supporters.