documentclass[a4paper,11pt,fleqn,twoside,openright]{memoir}
usepackage[danish]{babel}
usepackage{appendices}

begin{appendices}

    chapter{this needs to be called "Appendix A"}
        label{appA}

    [content of appendix A]

    chapter{this needs to be called "Appendix B"}
        label{appB}

    [content of appendix B]

end{appendices}

begin{appendices}

    chapter{this needs to be called "Bilag 1"}
        label{bil1}

    [content of bilag 1]

    chapter{this needs to be called "Bilag 2"}
        label{bil2}

    [content of bilag 2]

end{appendices}

documentclass[a4paper,11pt,fleqn,twoside
  %,openright
  ,openany% <- only for the example
  ]{memoir}
usepackage[danish]{babel}
renewcommand{restoreapp}{}% <- added

begin{document}

begin{appendices}
    renewcommandappendixname{Appendix}% <- added

    chapter{this needs to be called Appendix A}
        label{appA}

    [content of appendix A]

    chapter{this needs to be called Appendix B}
        label{appB}

    [content of appendix B]

end{appendices}

begin{appendices}

    chapter{this needs to be called Bilag 1}
        label{bil1}

    [content of bilag 1]

    chapter{this needs to be called Bilag 2}
        label{bil2}

    [content of bilag 2]

end{appendices}
end{document}