Let us consider a language that has both of those; let us further consider the language to have two reflexive markers, one for singular subjects, the other for plural subjects.
A further detail: the number congruence for the subject marker follows morphological number (at least almost all of the time), whereas the object marker follows semantic number (so, e.g. 'family', when speaking of the family as a bunch of individuals, will have plural congruence, but when speaking of it as an entity, will have singular congruence).
And the final piece of setup before we get to the thing I want to describe: there are two reflexive markers that can fill the object slot. One for singulars, the other for plurals.
(N.B. the language could conceivably also have duals, but they will not affect the detail I am about to describe, and so I will not mention them any further.)
Now, we can imagine that in some language, a group X having Y as a member can be expressed as 'X having Y'. For some contexts, this even works in English, so it shouldn't be particularly weird.
However, one could imagine that the particular construction mentioned there gets weird here:
X have-3pl-refl.1sg Y
and, one could even imagine, that this ignores the actual number of Y, that the (1/2/3)pl-refl.1sg affix on certain verbs simply signify 'have/acquire/... as a member or part'.
I considered working this idea into some Dagurib language, since those will, I think, have object congruence, ... however, with the pace at which my current conlangs are being developed, Dagurib might start getting done when I turn 120.