If this formal request is made by an attorney, then it is similar to a cease-and-desist letter (but not a cease-and-desist order, which in turn is similar to a einstweilige Verfügung [de] in German law).
Anyone who is harmed by the offending action and thus has an Unterlassungsanspruch [de] can send an Abmahnung.
Trademark violations, copyright infringement, and not complying with disclosure and information requirements harm the public, so the Gesetz gegen den unlauteren Wettbewerb [de] permits competitors, interest groups who operate on the same market as the offender, consumer protection organisations and Chambers of Commerce to file an Abmahnung.
Reimbursement of attorney's fees is excluded in cases of violations against information requirements, if the violation is committed on the internet and the filer is a competitor.
[2] Since the enormous proliferation of web sites, there have been waves of frivolous Abmahnungen, the so-called Abmahnwellen, as lawyers and copyright holders search for even minor and usually unintended violations of e.g. copyright law on the internet, with some law firms sending thousands of letters per year demanding payment.