The original bar is mounted in a plastic housing which i assume isolates it from the box.
Neutral wire bus bar.
When there is two neutral wires in one hole on the neutral bar of an electrical panel several things may happen.
Neutral bars are widely used in residential and commercial electrical service panels to terminate all those white neutral or return wires from the many load circuits in the building.
The black hot wires are protected from overload by circuit breakers in the panel which typically sense the current in the hot wire and trip out if current is exceeded for too long.
Information about what is a neutral bar and how it useful in service panels.
The white wire is neutral not ground.
The main feed is 3 wire with 2 hot legs to the main breaker and an aluminum wire to the neutral.
However the white and bare wires both have the same ground.
Your answer is yes you do put in the bare wire with the white wire.
The bare wire is a ground to.
There is nothing neutral about a neutral wire.
It is a current carrying conductor just like a hot wire and has all the potential for danger and should be treated with the same respect.
A double tapped neutral is when more than one neutral wire is fed into a single screw terminal on the neutral bus bar in the main electric panel.
The problem primarily comes from the inappropriately named neutral wire.
You can see this clearly in the picture below as there are multiple neutral wires feeding into a single screw in more than one instance in this spaghetti mess of wires.
They relate to safety and should be corrected.
Consider the schematic for a flash light.
If that voltage is not the same it will trip gfci.
Some gfci breakers have between 30 80volts on neutral.
There is also a bare ground wire from the neutral bar to the water pipes.