Bottom Bracket Height I am just wondering if BB height from the ground is the same for all bikes or for all mountain bikes, all road bikes, etc? Or does every brand have its own design for that dimension?
No, it is not. Different bikes in the same class (mountain, cross, road, etc.) may even have different bottom bracket heights even among comparably sized bikes. It is a per manufacturer design decision where to place the bottom bracket.

Mountain bikes need higher bottom brackets to clear obstacles vs road bikes or cross bikes, for example.