Welcome back to the Wefugees Community and thank you for sharing you question!
If you save money over months and years to finance your drivers license, I do not think, that the authorities will make any problems for you. I assume, that you also pay the training lessons etc. little by little right?
However, any additional income you might have next to the social money (as a Duldung holder, it would be the "Asylbewerberleistungen") needs to be reported to the authority (in your case probably the "Sozialamt").
Depending on the type of income and the amount, they will most probably pay less social money accordingly.
If you only saved the money from the social money, I don't see a reason, why the should stop paying and I also don't see a correlation to a potential deportation.
Would you agree, @MBEon-Sophie or @mbeon-bernd?
If a drivers license will help you to find a good job to cover your own (and your family's costs of living), it might open new perspectives for you in the future (i.e. through the so-called "Beschäftigungsduldung" or a residence permit according to §25b AufenthG).
I hope this helps and all the best,