When we were very young, we would apply this dressing to our rather scuffed shoes, by natural logic the inference was that the oil gave one a neat foot, hence the name.
But not so.
Though now so rare as to be called obsolete, neat was formerly a general term for cattle, more especially for oxen.
Neat’s-foot oil, accordingly, was and still is an oil extracted from the hoofs, now also the shin bones, of oxen or cattle.
Now used as a lubricant and for dressing leather, it was at one time also used medicinally.