There are many variants of the card came Rummy including Indian and contract Rummy. I can find no other reasonable explanation.

Well I believe this may explain the three and it is more likely the dart term came from a card game rather than a direct link to Shanghai. So, Shanghai has three counties, three of a kind, three within the same region. The 16 districts are Hangpu, Luwan, Changning, Putuo, Hongkou, Minhang, Jiading, Jinshan, Songjiang, Qingpu, Nanshi, Xuhui, Jing'an, Zhabei, Yangpu, Baoshan, and Pudong New Area. The 3 counties are Fengxian, Nanhui and Chongming. I believe the reason why Shanghai may be used is because Shanghai is divided into 3 counties and 16 districts. Although this explanation is plausible due to the fact a cribbage board was used as an early scoring device for darts, then 301 games. 'The origins of this term have been difficult to find, however, it could have originated from ‘Shanghai Rummy’ a card game when the object is to get three of a kind or three sets of three cards.

Shanghai card game