Given scene/plot descriptions from movies or TV shows, and given a specific question about the scene, summarize the question into a title

In Person of Interest, in "The Day The World Went Away", Finch's cover identity is exposed after he visited a cafe he brought Grace to. How did Samaritan establish the link between Professor Whistler and Harold Finch? As far as I know, Samaritan knows: A person who looks like Finch used to come to the cafe A person who looks like Finch used to order the same thing That, to me, is not enough to conclusively prove that Professor Whistler is indeed Harold Finch. What other information could Samaritan possibly possess to help it expose Finch's cover identity? Was it the fact that Finch was talking to The Machine?
How did Finch's cover identity get blown?