Abmahnung is not a proper lawsuit, it's a mostly legal way to get parties to agree in case of a legal dispute without having to actually involve a court. You can ask to be reimbursed for incurred costs for up to 1000€. The current government is working no reducing the abuse of this legal mechanism.
Sending one of these requires you also to clearly show how the recipient infringed on your rights or a law that affects you or your business in a negative way and lay out the costs you want reimbursed for sending the letter (up to 1000€). If you're wrong about it you'll have to pay all and any attorney fees of the other party.
If someone refuses to pay or sign the Abmahnung you have to go to court, in which case you'll need agreement from the party you're representing.
To file an actual lawsuit you still need to be wronged or damaged by the actions of another party, you can't just sue in the name of someone else without them agreeing to it. This is also limited to civil law, criminal law is handled by the state prosecutor via the police, not you yourself.