Inheritance has been around forever, probably longer than mankind — even animal offspring will probably inhabit burrows which once belonged to the parents. Hell, even a lactic acid bacteria would not hesitate to inhabit the same fold in the skin which its parent cell did.
It corresponds on a very fundamental level withto how we feel towards our children: We equip them in any way we can in order to help them be successful.1
This principle is so deeply ingrained that it is not bound to any particular social order. Even societies who are much less squeamish about taxation allow private inheritance, like the People's Republic of China. Monarchies did, obviously: "L'État, c'est moi" was Louis' XIV succinct way of describing the scope of his possessions which were then inherited to his great-grandson (because the precursors in the line of succession had died).
An attempt to abolish inheritance would be short-lived because it runs against our values; its proponents would swiftly be removed from office. That said, passing on wealth beyond what is needed for a jump start is considered unjust from a different angle (perpetuating wealth and influence, creating dynasties in a purported meritocracy). This is the reason why a taxation of inheritance, e.g. an estate tax, is tolerated. But the conflict with what we consider a natural right — to keep what is ours and to do with it what we want, in particular give it to our children — leads to constant resistance against and scrutiny of such laws. This articel by the Brookings Institution discusses this conflict.
1 That some parents think not spoiling their offspring is the best equipment notwithstanding ;-).