Welcome back to the Wefugees Community and thank you for reaching out to us again.
I assume your friend is holding a so-called "Duldung" now? In general, starting an "Ausbildung" is possible without a passport. However, you need to have the permission from the immigration office (Ausländerbehörde) as for any kind of paid job. In case your friend wants to apply for a so-called "Ausbildungsduldung" (Handbook Germany is providing very helpful general information on it in various languages here) it will be important that she proves that she did everything reasonable to clarify her identity - she should record her efforts very well.
I hope this helps and please don't hesitate to get back to us with any further questions.
All the best,