If there are two nearly identical copies of me in the same place, why is there no further interaction between them

Your two copies differ by states of many neurons, that's billions of particles. They are not "nearly identical".

It is tempting to think about "one different thought" or "one different perception" as very small changes. But on particle level those are huge changes. A small change on a particle level is something you can't notice, and therefore you can't notice as those copies of you interact... and when the small change becomes big enough, your copies are already decoherent.