>> What would be good customization options for that?  I imagine
>> a new boolean option 'isearch-buffer-local'.  Maybe later
>> it would require a new value to support even multiple isearch
>> sessions in different buffers.  In this case, all state variables
>> like isearch-cmds, isearch-lazy-highlight-overlays etc. could be made
>> buffer-local.
>
> I don't understand why `isearch-buffer-local' should be a user
> customization, since it doesn't change the external behavior of Isearch,
> but simply improves the implementation.  It might break third-party code
> that hacks into isearch.el, though.

Indeed, we need to care about third-party code as well.
So we could add a new option, then ask the users to enable it
to see what effect it has on user customizations, then after
the next release its default could be changed to t: