Welcome back to the Wefugees Community and thank you for reaching out to us.
First of all, the "Ausländerbehörde" (immigration office) is not responsible for the social benefits (Asylbewerberleistungen) for asylum seekers or tolerated (Duldung) people. Usually, the respective authority is called "Sozialamt".
If you are in Germany longer than 18 months, you may be entitled to receive the so-called "Analogleistungen" according to §2 AsylbLG, which are similar to the money people get from the Jobcenter. Otherwise you will receive the "normal" amount according to §3 AsylbLG or in the worst case the reduced one (sanctioned) according to §1a AsylbLG.
If I am not mistaken, and as I also took from this table, in case you are receiving the "Analogleistungen", you are allowed to own a personal capital of up to 5000 Euros as single household. If you own more, the Sozialamt will ask you to use it before applying for money. If you are receiving the allowance according to §3 or §1a, the respective amount for capital is only 200 Euros per member of the household/family.
In general, during the application procedure the Sozialamt will also ask for your bank statements of the last months. They will see if you withdraw a bigger amount of money and (as you mentioned already) investigate. Asking for social money even if you are not in need and you hide your personal money is considered as fraud and crime and I would highly recommend you not to get involved in those kind of matters.
Am I right on this or is there anything you may want to add dear @mbeon-fardeen?
However, please don't hesitate to get back to us with any further questions or if my explanations were not clear.
All the best,