So JKR briefed about the interacting pictures in the wizarding world in an interview as such
Q: All the paintings we have seen at Hogwarts are of dead people. They seem to be living through their portraits. How is this so? If there was a painting of Harry’s parents, would he be able to obtain advice from them?
JKR: ... They are all of dead people; they are not as fully realised as ghosts, as you have probably noticed. The place where you see them really talk is in Dumbledore’s office, primarily; the idea is that the previous headmasters and headmistresses leave behind a faint imprint of themselves. They leave their aura, almost, in the office and they can give some counsel to the present occupant, but it is not like being a ghost. They repeat catchphrases, almost. The portrait of Sirius’ mother is not a very 3D personality; she is not very fully realised. She repeats catchphrases that she had when she was alive. If Harry had a portrait of his parents it would not help him a great deal. ...
Now, my question is, why is the picture in the Muggle Prime Minister's office able to interact? How can a wizard's imprint be left in a muggle office?