Thank you for your question and Welcome to the Wefugees community!
A very similar question has already been asked and answered before, please have a look at the following thread from our platform: Is it possible to get a tourist visa to visit the US as a Syrian refugee who lives in Germany?
Since your boyfriend has already been granted asylum, he should be able to travel to the U.S. with the refugee passport, often known as "Blue passport" - but he will need a visa. His protection title would only be terminated if he traveled to his "homeland". Fact is: your boyfriend has a recognized residence title in Germany, so the process of applying for a tourist visa should be the same (at least I couldn't find sources stating something else).
However, I cannot say whether or not the application will be successful or how likely it is that he will receive said visa. You could try and reach out to a refugee council in your area in the U.S. and ask them if they know about rates of granted visas for refugees who applied for tourist visas.
Here, you can find information by the U.S. government on how to apply for a "nonimmigrant visa" (tourist visa): https://www.usa.gov/visas#item-213341
Also, if you are interested in getting information on How to Marry in Germany, please go ahead and use our search function (on the right!). Type in "marriage in Germany" "getting married" or the like, we have had plenty of cases that can be very helpful :-)
All the best,