I would like to comment on the British situation only, to not confuse various issues across countries.
In the British system it needs to be remembered that the person who is monarch functions in two ways - as the monarch and as a private person. Anything relating to the crown cannot be considered private. So items left to the crown are not for the person to treat as private property.
Property held by the Royal Collection is a third category, which the sovereign holds in trust, meaning the current monarch holds it for the next, and so on.
The Crown Jewels, referring to the regalia in the Tower, is a fourth category. It probably belongs to the State but pieces could belong to the Church. I think "ownership" pedants could debate these pieces endlessly regarding who they belong to.
I have long felt there is another category, particularly for the jewels worn by the Queen; jewels which are her private property but which pass down the reigning line of the family - perhaps under an informal understanding or by some formal arrangement these days, since Edward VIII. Royal wills are not made public and we can only conjecture. We have been aware of arrangments reached with governments over inheritances to avoid death duties - a practical measure to avoid dissipating a wonderful collection of private jewels that are used in a very public way.