When we touch the ground wire(!) on some of our power outlets, the tester lights up
That might be because you are using the cheapest (€1?) kind of tester - a screwdriver with a metal cap you have to touch.
Consider using a better sort of voltage tester that you can obtain relatively cheaply (€10?).
and the circuit breaker doesn't [trip]
A Circuit breaker only trips for high currents, those voltage testers only pass tiny currents (otherwise you'd be too dead to post a question here).
If your earth wires were really live, you'd be right to expect the circuit breakers to have tripped. Worst case is the earth wires are live and not properly connected to ground at the main distribution panel.
A better voltage tester would help you to establish the facts.
The copper wires in one of the distribution thingies ...>
A more usual name for those "distribution thingies" is "screw-terminal" "connector-block" or some variation.
... are discolored ... Is that an indicator of [arcing]?
In my (limited) experience, arcing usually produces much more pitting and blackening than is visible in your photo.
This discoloration could be a sign of overheating - though the plastic nearby seems unaffected.
However there should be no current flowing through the earth wires. That should only happen very briefly in a fault condition that quickly ends with the panel saving your house from burning down by tripping a breaker.
Usually, overheating is caused by not screwing down the connector tight enough.
Tightening the connectors ought to be sufficient - though I'd remove them and check for corrosion in the contact area. As a precaution you could replace the connectors with new ones of the same type and clip off the discoloured parts of the wires, re-trim the insulation to a suitable length and reconnect tightly.
A blue or blue-green colour can also be caused by some kinds of copper corrosion. I don't know what might have caused that in your photo.