People who came to Germany between January 1, 2017, and January 1, 2020, and intend to apply for a training allowance at some point must verify their identity by June 30, 2020. Meeting the deadline means that by then, the individual must have tried all feasible measures to verify their identity, even if the result of the identification process comes at some later point of time.
People who came to Germany after January 1, 2020, must have their identities verified within their first six months in Germany.
Those who came to Germany before December 31, 2016, have to prove their identity when applying for a tolerated stay permit ("Duldung").
Even if you miss the deadline or in case, despite all your efforts, you have not been able to prove your identity; you may still be able to obtain an “Ausbildungsduldung”. But you are not legally entitled to it- which means Immigration Office' decide whether to issue you the permission.
Important: You have to document your efforts for verifying your identity. So if you, e.g. go to your embassy to ask for an ID, ask them to confirm in writing that you were there. You can also take photos, etc. The more evidence you have, the better. If you are concerned you may get deported, seek advice from a counselling centre or a lawyer in advance. You can find a counselling centre nearby, e.g. on proasyl.de. To search for a lawyer, check rechtberaterkonferenz.de.