How can I determine the style, design, and manufacturer of this tile?
This will require a lot of leg work, and some phone calls.

The only way to match it up is to go to the tile supply houses in the area and physically compare it to the tile that is in their inventory or samples to order by.
That should be your best shot at it. 

While at the various supply houses, I don't mean the big box stores, but don't count them out either for this, there may be a knowledgeable person there that may tell you, "Oh yeah, this looks like such and such of tile, check the store in (fill in name or place here)"