The fundamental answer is that the billionaire does NOT own people or sovereign power, just property.
In case of the King, the King got to choose who would inherit the sovereign power. With power, came property (including in some cases the kingdom's subjects), but the power - the right to rule - was what was inherited.
In case of the billionaire, what is inherited is property. Same as you yourself, when you die, if you own a computer and some money in bank accounts, have a right to will them to your children or whoever else you wish, same with the billionaire - the only difference is the number of zeros on the value of the property.
But fundamentally, there's no difference between the inheritance of those $30B USD and inheritance of $X USD worth of assets your heirs will inherit from you - and both have nothing to do with inheriting sovereign power as the King's heir would.
As a side note, don't confuse money with power. Barak Obama was born reasonably poor as far as I'm aware, and is now arguably the most powerful man on Earth (and if you count the nuclear football, more powerful than pretty much any King who ever ruled). Money may ease access to power in some circumstances, but it does not in and out of itself grant you power by virtue of having the assets (ask the people with money murdered by Bolsheviks between 1917 and 1925 about that if you don't think that's the case). As Mao Zedong wisely noted, real power only comes from the barrel of a gun; or as a significantly wiser Ahiqar said, from the word able to sway other people's opinions and actions.
Minor update: having re-read the answer, it covers the main political angle, but omits the minor legal one: Feudal monarchy inheritance largely followed very rigid and well-established rules (often, Primogeniture). Modern property inheritance is - largely, at least in USA - arbitrary given a valid will exists and the legal challenges to the will don't succeed. Bill Gates can write out all his descendants out of his will and give 99.99% of his property to Malaria Relief Fund, fully legally.