Like a second shot at it

I agree, in everyday usage, two bites at the apple means two chances to do the same thing, so a second shot at something. You also indicated that you thought there was a legal term "two bites at the apple" and there is a concept with this name. It means that a plaintiff (the person suing the defendant) in a civil or criminal proceeding should not have the ability to sue that same defendant a second time with the same underlying facts of the case, even if they try to do so under a different legal theory. You need to use all of your legal theories the first time. In addition, you cannot try to sue a new defendant with the same exact facts after you lost. You should have included them in the first case and you already lost. There is case law about this in the United States. I am not a lawyer, so I cannot say if two bites at the apple is legally binding throughout the United States.

have (or make) two bites at a cherry = have two bites at the apple Let us do this work at once and not make two bites at the apple (we will not do this work so long).

There,Two territories to each other, struggle to compete to.