<p>Why do Linux distros have different package managers?</p>

<p>I find this very strange because other software such as text editors, desktop environments and graphics software (Inkscape, Blender, GIMP) are shared among distributions. Why not have a common setup tool?</p>

There are also many different text-editors, desktop-environments and so on. The different distributions share this only, because they provide all these programs.
But they have to decide for one package-manager. Different package-manager wouldn't know about software installed by another package-manager. So distributions decide for one or develop one themself, tailored to their specific needs.
Two very common package managers are RPM and apt, that are both used by different distributions.