Why did everyone keep saying to Harry that his eyes are like Lily's?

When people say you take after a parent, they might mean it in looks as well as character. When they say you have their eyes, though, it might suggest something more. The eyes are the only organ directly connected to the brain and are said to be the window to the soul; they convey many emotions and thus could imply a person's deeper nature.

James Potter was a pure-blood, a Quiddich star and a reckless and sometimes arrogant student. Lily Evans, however, was a muggle-born, kind and a very good student according to Horace Slughorn. So when people keep saying Harry looks like his father but has his mother's eyes, I think they mean Harry's general behavior and his deeper nature. To the outside observer, Harry is a Quiddich star and an OK student who's sometimes prone to recklessness and anger. But deep inside, Harry is also humble, resourceful and good-hearted, like his mother was as a student.

From The Deathly Hallows:

“—mediocre, arrogant as his father, a determined rule-breaker, delighted to find himself famous, attention-seeking and impertinent—”

“You see what you expect to see, Severus,” said Dumbledore, without raising his eyes from a copy of Transfiguration Today. “Other teachers report that the boy is modest, likeable, and reasonably talented. Personally, I find him an engaging child.” [...]

“He is his father over again—”

“In looks, perhaps, but his deepest nature is much more like his mother’s."

And this might be the key to understanding his relationship with Snape, and Snape's loyalty which is eventually one of the more important elements of the series. Snape mostly despises Harry who looks so much like his father, the man who bullied Snape and then snatched the woman he loved from him. But Snape agrees to keep Harry alive no matter what because Harry is the last bit he has left of Lily. His eyes also remind Snape, a former Death Eater, that his greatest love was a human ‘mudblood’. This is why he wants Harry to look at him before he dies, to remind him of the soul he did it all for.

It's a very normal thing that people say when they meet children of those that they used to know very closely. Pointing out physical (and especially facial) similarities in a child with its parents is something lot of people do, most probably out of this slight amazement that how nature (technically genetics) works in such a marvellous way that people transfer some of their traits almost exactly to their kids.

The expression might be coming out of 2 main reasons:

1) Happiness to once again see a glimpse of one's friend or relative in their child (especially when the said parent is deceased which is the case here in HP movies). And,

2) As mentioned above, (though very well known and understood but still) an appreciation over how beautifully and meticulously nature works through genetics.

If someone has a physical feature that is rarer than usual, such as green eyes, it can be an involuntary reminder of someone. My sister has the same hair as our father had and it's her most unique feature but also tinged with a little sadness that he isn't around. It's like how the Weasleys are redhead that's a rarer feature and if you can tell which parent they're from it's a nice connection. Green eyes are the rarest and I believe Lily's and Harry's eyes were exceptionally so green, Petunia doesn't have them they were very much Lily's. The movies fucked up and didn't even have the actress playing Lily have the same eye colour as Daniel Radcliffe so in the movies it makes zero sense.