On 'Studio 60', the lawyer who was representing NBS and the late night comedy show 'Studio 60 Live On The Sunset Strip' against charges of sexual harrassment worked for the law firm of Gage Whitney. Sam Seaborn had been affiliated with the same firm when he wasn't working in 'The West Wing'.
But this doesn't link the shows to the same TV dimension. During the run of 'The West Wing', the President was Jed Bartlet, while it was Bill Clinton and George W. Bush in the main Toobworld (as it is in the real world).
However, it does serve as a good example that it's not just characters which can have counterparts in several different TV dimensions.