It is difficult to see how your boyfriend could have an asylum case processed in Germany after being rejected in Sweden. He could lodge what’s called a secondary application (in German ‘Zweitantrag’). However, for this to be admissible, there would have to be substantial new evidence in his case or changed circumstances (for example in the law or in his home country). Here is detailed information about the possibilities open to people in Germany who have had a frist asylum case rejected:
As for you, it is not clear where you are in the EU. If you have a visa for the EU country you are in, that country may be responsible for any new asylum application you make. I am not aware of what influence the rejected application from 2020 would have.
A question from me would be: if your aim is to get an Ausbildungsduldung, why not apply for a regular residence permit for the purposes of doing an Ausbildung?
Here is the information from the German embassy in Sweden about applying for a visa to do an Ausbildung. The procedure would be largely the same in all German embassies: