I should start of by pointing out that when we talk about a "chassis ground" we usually mean we are connecting the chassis to some larger and better established ground potential like the earth itself. The reason to do this is for safety. If a high-voltage wire inside the equipment gets loose and contacts chassis, you'd rather the current be diverted to ground and a breaker gets thrown, than for the chassis to be sitting at high voltage waiting to electrocute the next user who touches it.
You are talking (sort of) about connecting the circuit ground to the chassis, which is not the usual meaning of "chassis ground".
Now if I connect the anode to the chassis, electrons flow from the anode to the chassis until both have the same charge density, right?
Charge will flow in a conductive structure until the two parts are at equal potentials.
Charge density will not be uniform throughout the structure. Charge will tend to accumulate near where there are external charges of opposite polarity, and will be depleted near where there are external charges of the same polarity.
And if I connect a load to the battery, electrons from the chassis and the anode are exchanged such that the charge density remains approximately the same?
Electrons are exchanged until the potential is equal between the battery's anode and the chassis.
So could I say that after chassis grounding, I have the same kathode as before but a bigger anode which consists of the former anode and the chassis?
Yes, that's reasonable.
If the whole chassis gets charged, isn't there a problem that the charges may somehow escape to the environment?
No. Kirchoff's current law says that current only flows in completed circuits. Charge may find a path back to the battery's cathode, and that would tend to deplete the battery. But it won't just flow away "into the environment" without the environment being connected back to the battery's cathode.
Do we have to pay attention to isolate the chassis from the environment?
how the potential can be equal if the charge density isn't.
See the electrostatic Poisson equation:
In fact, if the charge density is non-zero, then the potential will be nonuniform.
Only when the net charge density is zero (which I guess is one example of a uniform charge density) will the potential be uniform.
In a condutor, the static behavior is for the free charge to move to the surfaces of the material, leaving zero net charge and a uniform potential within the bulk of the material.