Does "no backing out" have the same meaning as "no backsies"? Could you teach me how to use this expression in polite way?
Yes you are correct. Originally "no backsies" was a phrase used by children (usually) when playing a game or engaging in some sort of argument. It basically means "you can't do to me what I just did to you." However it has changed to because a phrase which means that you can’t take it back or you can't back out. So if someone said something and tried to take it back you could say “no backsies” saying “you can’t take it back”. This is a very casual phrase, and is still seen as kind of childish. So I would not use it in formal or serious conversation. The phrase itself however is not impolite.