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The Department for Foreign Affairs (Auswärtiges Amt) states the following:
In order to register a marriage that was made in another country in Germany you need to provide your local registry office (Standesamt) the certificate of marriage. The office will then check for its authenticity. In addition they might ask for a so called "Apostille". This means that the Danish state gives you a document that attests that your certificate is valid.
There is agreements between Germany and other states that they don't have to issue such an "Apostille". Denmark is one of those states, but I don't know if certificates of marriage fall under the agreement or not. I would suggest to ask your local Standesamt about it.
Keep in mind that you will probably have to hand in a translation of the certificate that was done by a certified translator as well. If you submit a translation that was made abroad they might not accept it. Once again: it varies from office to office. The best way to find out how cases like that are handled in your Standesamt is by asking them in person :)
Source (German language): https://www.auswaertiges-amt.de/DE/Infoservice/FAQ/Eheschliessung/05-Welche%20Unterlagen%20-%20Ausl%20Heirat%20in%20D%20registrieren.html?nn=332718
I hope this answer was helpful to you. If not, feel free to ask further questions!