I think this should achieve your goal

let some_on_click_handler = Callback::new(move |evt: FretClickEvent| {
    if let Some(node) = error_text_node_ref.get() {
      // access the current style
      let style = window().get_computed_style(&node).unwrap().unwrap();
      let animation_name: String = style.get_property_value("animation-name").unwrap();

      // [...] do something with this information

      // change the style
      node.style("animation-name: shake");
    }

    // some other logic for setting error_text..
});

First, you get all the current styling of an element using Window's get_computed_style() (Rust binding by web_sys of the JS function) and then use get_property_value() to access a specific CSS style value.

Then, to set the style you use the style() which you found yourself. You can e.g. pass it a dynamically created String.

instead I want to access the style and change one property on it

What you're asking for here is mutable reference to a specific style, and this is impossible as far as I know.