As for how tartans came to be associated with different families/clans, there is debate about how this came about and it is actually fairly complicated, but to make a long story short...
First people made their own tartans in their homes. The patterns were whatever that person wanted to make. Later as people started having more specific jobs and traded or bought from others, then there would usually be one fabric maker in an area. This fabric maker would make whatever tartan they liked.
People in the area were pretty much stuck with that pattern. In this way the pattern became a regional tartan. Since clans would be associated with regions, these regional tartans became clan tartans.

As far as making your own tartan and getting it registered goes everyone from American Express to the Katsushika Scottish Country Dancers of Tokyo have their own tartan these days. If Canadian provinces, American armed forces and Madonna can have their own tartans made and registered, you should be able to also.