Sure, you can. You can even build tubs from tree stumps and wooden boats, and solid rock, and marble, and Granite, and iron, and steel, and plastic and fiberglass. The design and fabrication is kept to the standards that are readily available, so that they can be readily mass produced, and therefore manufactured in volume, and sold competitively. The secret is in weight, construction costs, volume, thermal retention, expansion and stress fracturing, surface destruction from soap and scouring, water retention and leak free supply and drainage, slip and fall hazard, and next home buyer acceptance. If you carved out a solid heart redwood or Sequoia log and polished it beautifully, and perfectly machined the drain hole, you'd have a tub the neighbors would love to party in.
: I seen pictures of tubs made from tile and slate. Is it possible to build these your self?