But if it takes you apart then how does it know how to put you back together? It has to follow a blueprint of some kind otherwise all you have is matter, not living matter. So before it takes you apart it must make a complete recording of your molecular structure. If that recording is then used more than once then you end up with copies of yourself.
Unless some intrinsically part of that matter stream is what amounts to your soul. That spark of life. Trek implies they can move that about even without knowing what it actually is, and they definitely refer to its definitive existence (Katra, the various but relatively few NDEs, the times characters spontaneously evolve, including VGer. Vger manages to copy it without even meaning to, because it's beyond its understanding, in the Ilia probe. Ira Graves may even have housed it in Data, and various androids seem to have developed something like one.)