'Elementary' takes place in the TV dimension in which Sir Arthur Conan Doyle never wrote the Sherlock Holmes stories back in the Victorian Age. (It's also the same dimension in which Washington Irving never wrote "The Legend Of Sleepy Hollow".) 'Person Of Interest' takes place in the world of Earth Prime-Time. Now, it could be that the characters from each show do have dopplegangers in the other's TV dimension, but I think the easiest splainin for this confrontation is that it takes place in the Promoverse.
USA Network used to be a master at making these promotional commercials for their shows. The characters from 'Burn Notice', Psych', 'Royal Pains', 'Monk', 'The Dead Zone', etc, have all interacted with each other in these promos. Many of them could have been pozz'ble, just pozz'ble, in Earth Prime-Time, but some just have to be considered only to be found in the Promoverse. For instance: the interaction between 'Monk' and 'The 4400', and 'Psych' with 'Defiance'.
All of the entries for the Promoverse may just be small offerings, but they promise a whole world behind them.....