His life was at stake, then by Allah's permission, he was saved. Now he wants to sacrifice animal to please Allah because he believes it is recommended in Islam to sacrifice a life in exchange of the favor Allah bestowed upon him saving his life. Is this something from South Asian Islamic culture or there are authentic sources where Allah/Messenger (S) prescribed so?
The reason behind sacrificing animals(Specially a Lamb) to return a blessing that you get from God is not "to sacrifice a life in exchange of the favor", It is about thanking God by helping poor people and giving them meat(a nice and proper meal that was hard to afford).
So the main purpose of any such sacrifice in Islam is just like charity not the death of that animal or .... (I've heard even some people give the meat of sacrifice only to their own family who are not poor at all),
Any deed of such kind with a purpose other than pure charity is considered at least by me, as a cultural or self-made belief not necessarily wrong but not related to Islam.
Allah has mentioned in Quran that no soul will bear the bourdon of another. Also one person will not be asked about the deeds of another. The sacrifice done by Muslims on Eid Ul Azha is to remember the sacrifice of Prophet Abraham (PBUH) of his son that Allah accepted and replaced his son with a lamb