how nice that you turned out to WeFugees with your question.
Every country has its own regulations when it comes to necessary documents for marriage. Of course you have the possibility to get married here in Germany. You do not have to be registered in Germany for 21 days to do so. After the marriage, you must have the marriage legally recognised or registered in Greece.
If you marry an EU citizen, your spouse can travel directly to Greece with you without a spouse visa. A third-country national who has a residence title in another EU country (Germany) can also travel to Greece with you without a visa according to § 2 para. 4 of the Freedom of Movement Act/EU.
But now you say that your partner has a Duldung for education. A Duldung is not a residence title. It means that someone actually has to leave the country, but is allowed to stay in Germany for the purpose of training. Of course, you can still get married - in Germany or Greece, as long as you have the necessary documents. However, your husband, as a third-country national and holder of a training permit, may then need a spouse visa before he can join you in Greece. In this case, one more administrative step (application for a visa) would be necessary after the marriage.
In order to marry here, various papers must be presented: Usually a passport, birth certificate or other proof of parentage, extended registration certificate, a "certificate of marriageability" (certificate stating that you are capable of marrying according to the law of your country of origin, especially that you are not yet married or no longer married) and other papers depending on the law of the country of origin. Papers from some countries of origin must also be checked for authenticity by the German Embassy. You can get detailed information about the required documents at the competent registry office where you want to get married.
If you decide to get married in Greece and you want your husband to be allowed to stay there immediately, you will usually have to present further documents before the wedding. In principle, the administrative act just described (application for spousal reunification) is applied before the wedding and not afterwards. More detailed information on this can be obtained from the relevant registry office in Greece. These are documents such as a visa application from the Foreign Office, passport/identity card, proof of health insurance cover, proof of sufficient A1 language skills and an informal invitation from the family member already living in Greece.
I hope my explanations are of some help to you. I wish you and your partner all the best.