TOKYO— Yoshiaki Fujimori wants to be the Steve Jobs of toilets.

Like iPhones, app-packed commodes are objects of desire in Mr. Fujimori’s Japan. The lids lift automatically. The seats heat up. Built-in bidets make cleanup a breeze. Some of them even sync with users’ smartphones via Bluetooth so that they can program their preferences and play their favorite music through speakers built into the bowl.