OP, we are like you -- we read a lot of older classics to our kids but let them choose what they read for pleasure with few limits.
I like it this way because kids no longer get exposed to a lot of literary classics in school. And I definitely think it is impacting their taste in reading material, as well as their personalities. One of my daughters is very obsessed with the Narnia books and I think it's become a core part of her personality. She loves fantasy literature in general but is particularly drawn to the morality of the Narnia books (we are not religious but we're not anti-religious -- DH and I were both raised Catholic but are now agnostic).
I do love how these books expand their vocabularies and teach them about other times and give them some connection to the past. We also read the Little House books with them and that had a big impact on them. I remember my youngest daughter was extremely affected by a scene in those books where Pa punishes Laura by spanking her (my kids have never been spanked and did not know that was something that could happen -- it opened up a conversation about how DH and I were both spanked as kids and how we felt about it and why we chose to raise them differently). Those books also prompted conversations about the taking of Native land in the United States, and also how racism and prejudice manifests in different people (Ma, who also has good qualities, is very racist toward the Native characters, whereas Pa and Laura are not overtly racist though Pa still feels entitlement to the land). I like that reading older books often prompts difficult conversations, and that my kids are reading about people who don't always have the same values or ideas about the world. A lot of contemporary fiction is very sanitized and designed to smooth over these conversations. It is easier but in the long run I don't think it's good for kids. Fiction is a great way for children to learn about the world beyond their experience because it's not as risky or scary as encountering these things in real life.
Anyway, I was absolutely shaped by the books I read as a child and I am definitely using the books we read to our kids to put certain kinds of influences and ideas in their lives.