Is mentioning the names of friends while complaining about problems that occur with them, in order to identify the loyal friend from others, considered a form of backbiting?

If the narration is intended to warn against those whose company is impermissible, then it is permissible, provided it is limited to what is necessary. Otherwise, it is forbidden, because the default ruling on backbiting is prohibition. Backbiting is permissible for a sound, legitimate purpose that cannot be achieved except through it. This includes warning Muslims against evil and advising them, such as discrediting narrators, witnesses, and authors, or warning against those who undertake giving fatwas or teaching the Qur’an despite lacking qualification, or being فاسق (transgressing), or being an innovator. The condition for this is that it must be for the purpose of offering sincere advice for the sake of Allah Almighty, without any other motive.