Is it also possible to add the rights for closing a POS session for a user with the pos rights User?

I think this needs a bit of clarification.  In Odoo, we need to seperate "named users" (who pay a monthly user fee), and "employees" of an organization.  Employees interact with Odoo (say for example, a POS cashier), but do not necessarily "use" Odoo (say for example, closing a session).  You can have unlimited free employees, and the number of named licenses users is limited only by your budget.  In the POS, the default user (admin) will always be shown in the POS session login screen, together with any additional employees you have chosen in the POS settings "authorized employees" tab. Employees and "named users" are linked via the HR settings tag "related user."  This is a very important Odoo concept. If a restaurant does NOT want to add an additional user (cost) they can just appoint an "employee" to login to the POS session under the default admin PIN.  They can then close a session. This saves both cost, and allows for limited admin authority in the POS session.  I hope I'm clear.   

here your exact answer

that cashier should not be in Authorized Employees 

In POS , whoever you select employee for changing cashier , that cashier(Employee) must be have related user (user) and most important it must be manager (administrator) 

for this you must follow this flow: go to User > search for which user you want to put as cashier > then scroll down >there is option called point of sale > select as administrator.

this you can see or understand by following below code:

get headerButtonIsShown() {
    return !this.env.pos.config.module_pos_hr || this.env.pos.get('cashier').role == 'manager' || this.env.pos.get_cashier().user_id[0] === this.env.pos.user.id;
}

above mentioned that cashier must be manager

thanks