DOC: Update merge copy parameter code and doc

[liang3zy22]

• closes Give the merge copy parameter better documentation #4599 (Replace xxxx with the GitHub issue number)
• Tests added and passed if fixing a bug or adding a new feature
• All code checks passed.
• Added type annotations to new arguments/methods/functions.
• Added an entry in the latest doc/source/whatsnew/vX.X.X.rst file if fixing a bug or adding a new feature.

[mroeschke]

Not sure if this description clarifies when a copy is needed

[liang3zy22]

I have checked merge copy related codes, found that following codes will decide if copy:

pandas/pandas/core/internals/concat.py:229
        if needs_copy and not indexers:
            mgr = mgr.copy()

The needs_copy parameter is defined by user input, CoW status and concat axis.The indexers parameter is defined by following codes:

pandas/pandas/core/reshape/concat.py:666
            for obj in self.objs:
                indexers = {}
                for ax, new_labels in enumerate(self.new_axes):
                    # ::-1 to convert BlockManager ax to DataFrame ax
                    if ax == self.bm_axis:
                        # Suppress reindexing on concat axis
                        continue

                    # 1-ax to convert BlockManager axis to DataFrame axis
                    obj_labels = obj.axes[1 - ax]
                    if not new_labels.equals(obj_labels):
                        indexers[ax] = obj_labels.get_indexer(new_labels)

                mgrs_indexers.append((obj._mgr, indexers))

I don't know why new_labels can be not equal to obj_labels. They are both from objs, although there are some operations in self.new_axes function. I have tried several examples, but can't get an indexers value which is not None.

[mroeschke]

Thanks for the pull request, but it appears to have gone stale. If interested in continuing, please merge in the main branch, address any review comments and/or failing tests, and we can reopen.