Does the origin of an object always have to be at zero on a number line or x-y plane? Why or why not?
If you are asking this in relation to a Physics problem involving motion (for example), the answer is no. Generally, you can designate any convenient spot to be the origin of the motion.
As an example, if you are dealing with a falling object that is dropped from the top of a building and is caught by someone leaning out the fourth floor window, it makes sense to call the "origin" the top floor of the building.
In another situation, when you deal with potential energy, there is no absolute "origin" that needs to be designated. You assign the value of zero to any convenient location that makes sense - even if it is infinitely far away (as is the case in problems involving satellite motion).
I'm not certain this is the context of your question! Hope it helps.
(A mathematician might give you a quite different answer!)