I found this mention about Lucky Star in an anime blog.
it’s completely believable that they are a group of friends that appreciate each other’s company. I believe part of what makes this achievable is the ease and fluidity in which the characters are set up
When I read that, the reason I like Lucky Star so much suddenly all clicked into place.

The strange thing about Lucky Star is that the character's relationships with each other seem utterly natural. More so than any other school slice of life series, Azumanga Diaoh included. In Azumanga, you never really had a sense that the girls had individual personalities, they were just always doing things together, and they only had a few sparse things to distinguish from each other. Yomi wanted to diet, Sasaki liked cute things, Osaka was an airhead, etc. You never really seemed to get a sense of anything else about them. There was a exaggerated personality trait for each girl, but that's all they really were. Then they applied random other wacky characters (Kimura) and put them in fairly awkward and occasionally hard to believe situations for "slice-of-life." It certainly makes for a more lasting impression, but over time, its shock factor no longer shown as brightly in my memories.

In Lucky Star, they are all compressed collections of moe traits, but that's only to drag the viewers in. If you move all the moe aside, we get a very clear picture of each of the girls (except Miyuki) because we see them at home, by themselves, and all they ever do is talk about things that interest them. They're very distinct from each other.

The Hiiragi twins are very different from each other, and have very different interests. Kagami even tries to press some of her own interests onto Konata, thinking they might have something more in common. Tsukasa is more down to earth, will very likely end up a housewife, and just enjoys life. She's the perfect counterbalance between Kagami and Konata. Konata's personality is very obvious, I'll save space instead of elaborating. Surprisingly, even all the first years have fairly well developed personalities after just a few scenes in episode 18. Hiyori is into doujin circles, and if you've ever known someone like that in an anime club, they share a tendency to be ashamed of their hobby. In 19, she's just like any art student I've ever known, very attached to her favorite tools. Patricia "Bouncy-tan in the OP" is the Embrace-everything-"Japanese" exchange student, including all the character traits you see in any real world anime convention attendee that goes to Japan. They "know all about" all sorts of key spots, but do the cliche things, and ignore other landmarks. Iwasaki has a cold demeanor, but is deeply caring, and worries a lot about how she appears and is viewed. A typical person with lower self-esteem, they tend to care more about others.

They are written more like people than characters.

The way they interact comes off more natural too. People with very different interests gradually become tight-knit friends, especially when they've been together in school for so long. They aren't just randomly attached to each other. But neither are they exclusively friends with just each other. Miyuki is always off doing something or gets left out, Kagami has friends in her own class that she doesn't share with her twin. So far, we've never seen Tsukasa even talk with Misao or Ayano. Tsukasa is much closer to just Konata, while Kagami puts in effort to spend time with her own unique subset of friends.

They come off more realistic in their personalities and interactions than most other series. For a show that's all anime references, it's strange, but very nice.