To be more precise , the French assurance chomage is not exactly given by the government. It has been created by the government in 1958 and it has general interest goals, but the "assurance chomage" is managed by a non profit institution that collects contributions of all the workers and their employers.
It is not a public system with funds coming from national taxes but a compulsory collective system. As soon as you begin to work, a part of your salary is taken, and you employer must pay a contribution. All this money is given to an institution that collects all the money and provide then alocation to the job-seekers when they have lost their job.
Dirigeants of this institution are not Secretaries or member of the national public administration, but representants of the major trade unions, and this instution has got its own administration (the members are not Sate public servants).
In brief it is called "assurance" because it works like an insurance: the contribution you and your employer give is collected and is used to cover the risks (loss of money) of those that lost their job. And since you have contributed, if one day you loose your job, you should receive a granting.
The difference between an insurance is that
-there is only one institution and belonging (contributing) to it is compulsory for every worker and employer,
- this system is not meant to make money like private insurances (actually it is in deficit and there are taxes and public funds in addition to the contributions to make it work), but it has been created in order to enable every worker to be covered in case they lose their job. The percentage of contributions and the risks that are covered are the same for everyone.
I hope that it brings some more precision without making confusion.