Hi Qray,
I can understand this question in two ways, that the axis stays in relation to the given object points, or that the axis stays, but the points are lined up (moved etc). You can always “store” the information of an object axis with a Null, then align all the points as mentioned below and use the Axis tool to move the axis back.
Since you have provided no scene file, it is not really easy to answer in a professional way, I have to assume too much.
Typically I would suggest to work with Nulls as parent, and leave the object axis alone! For many reasons, but certainly for anything that involves animation. The point values of an object are ALL related to this axis. The Axis is the world to the points. Any change after animation to points is give, might produce trouble.
The direct way to move an object with the same structure (as in absolute exact match) would be via Xpresso and point iteration.
There is no option in the Axis Center to determine an axis based on a different object, not to define the axis outside of the points.
Why not create a copy of one of the objects and then move the axis of the copy where the “other” object originally has its axis. Done.
There are different ways to align it (e.g., via “Align Grid to Selection”, then use a Null, etc), but none of those options is just a simple click and go. the safest way would be to position three nulls on each object, each time on three points of the mesh. Null 1a, 2a, 3a, and 1b,2b, 3b. The Null 1 after positions is the “Parent” to each object a or b, the second and third null is used then for the Target Tag (attached to Null 1) as target as well as up vector.
When this is done, delete the Target of Object B, then Null 1b is placed under Null1a, and Rest PSR is used once (or Postion and Rotation is set to zero manually). I’m not aware of any native tool that can do this more precise. (Which points to chose? Just a nice triangle, so it’s stable)
Screen capture (you can download this clip for a better viewing experience)
https://www.amazon.com/clouddrive/share/64zLYF1DOSCq4OA0Lg39rgOeyQJhugDGaPINAZIpizz
To keep the Axis’ “home” memorized, please have a look here: screen capture
https://www.amazon.com/clouddrive/share/HBg2L8LxR4u2VELjuYeAIK6Hf0yBrrH97FPXdqDeIjj
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All the best