Welcome back to the Wefugees Community and thank you for reaching out to us again.
Regarding your first question, I don't see any reason why it shouldn't be allowed for a couple to make an "Ausbildung" at the same employer.
Regarding your question, if the Ausländerbehörde will approve your Ausbildung: This is dependent on a few factors and I would highly recommend you to contact the Ausländerbehörde (immigration office) directly. People holding an Aufenthaltsgestattung (pending asylum procedure) have a waiting period of three months before they are allowed to work. Since you are in Germany for more than 2 years, I assume that this period will be over at least. For more information please also check the official website of the Agentur für Arbeit.
As long as you are in the pending asylum procedure, the immigration office will not ask for your passport as you are also not allowed to visit your country of origin's embassy, for instance. In case your asylum application got rejected, they might ask you to organise a passport. Then, it will be important to prove them your serious intentions to obtain one.
If you want to move to another place during your pending asylum procedure, you need to submit a so-called "Umverteilungsantrag" to the immigration office. As far as I know, they will forward it to the immigration office which would be responsible for you after you moved. You will need to wait for an approval.
During the pending asylum procedure, people are in most of the cases not entitled to receive "Berufsausbildungsbeihilfe" (BAB) - a financial support for people enrolled in an "Ausbildung". However, you may still receive "Asylbewerberleistungen" (asylum seekers allowance) from the local "Sozialamt" in case your income is not sufficient to cover the life expenses of you and your family. See also this publication (in German though) for more details.
Am I right with my explanations and/or is there anything you would like to add, dear @mbeon-Christine?
I hope this helps and please don't hesitate to get back to us with any further questions.
All the best,