My personal experience is that even with a name like Young people ask if you're related to someone they know, but not everyone does, so perhaps it just never occurred to this bright young doctor.

She rushes to the lake house mailbox and places in it a letter telling Alex not to come looking for her in Daley Plaza. He will, however, keep them in mind when he is preparing to leave the lake house, and this will confirm the connection that causes her to take the house when she goes to Madison for her residency. Yet on this particular day she steps on the rug and realizes that the floor is loose beneath it. Even that is not a problem, as it simply means we have interlocking N-jumps:  in the first version of the letter she gave a different address, but when history moved her to Riviera it happened before that first letter was written, and that timeline is repeated with the new address, triggering a repeat of all intervening timelines. In this regard, it is fortunate that Alex died at Daley Plaza. The trees are in the back of his pickup truck, and he proceeds to plant them, but for one. We can suggest that the filmmakers didn't think of that, and it really should have been the younger tree that Kate got; or we can argue that since Alex intended to give Kate an older tree when he sent a younger one it arrived older as part of the magic; or we can imagine that when he moved the one tree that had originally been planted at the lake house, two years later that move caught up with Kate and the older tree vanished from the lake house and appeared in the city. Having concluded that it is impossible under fixed time theory, I am addressing what happens under replacement theory. At the end, we saw Kate at the lake house as Alex drove up - he got out and walked over to Kate who said 'You waited.' Now she is undoing that. He in 2006 is rushing to the lake house to check her old letters to confirm time and place so he can find her. They embrace. Kate had named her dog, who was female, Jack - and she didn't know why. It is one thing to say that the book, like the tree and the grafitti and the letters, leapt forward two years to appear in Kate's bedroom at that time. His father built that house, but he as a child apparently helped and lived there with his parents for a time. to the quiz link under the stated answer and then click "Report error" at the bottom of that page. She wrote Alex a letter telling him to wait two years and not to cross the road to meet her.

Edit, In 2004, after he finds Persuasion at the train station, Alex writes to Kate and says, "Kate, I have it with me and I'll find a way to get it to you sometime." In September 2004, now fully aware of Kate, Alex attends her birthday party, and they dance and kiss, but she doesn't yet know who he is (remember, she doesn't become aware of him until 2006).
This is plausible, and launches our timelines for us.
It is much more difficult to find a reason for Alex to be killed there, but if he is not that creates problems for several later events in the film. So the big problems are that if he lives the entire story becomes impossible, in multiple ways. Thus there is sense to this connection to a stranger triggering the magic, starting the story. Then they kissed for the first time. Kate and Alex both worked in Chicago, and they both had connections to the house on the lake. Kate was wearing a long red coat, and Alex was wearing a long brown coat with the light brown turtleneck underneath. Undoubtedly she has been in that room almost every day since she left the lake house, and it has by now been over a year. A significant shift occurs the day he realizes that there is a temporal anomaly involved.