Another option is to inject a short js script with st.components.v1.html
to do the styling:
import streamlit as st
def color_selectbox(n_element:int, color:str):
js = f'''
<script>
// Find all the selectboxes
var selectboxes = window.parent.document.getElementsByClassName("stSelectbox");
// Select one of them
var selectbox = selectboxes[{n_element}];
// Select only the selection div
var selectregion = selectbox.querySelector('[data-baseweb="select"]');
// Modify the color
selectregion.style.backgroundColor = '{color}';
</script>
'''
st.components.v1.html(js, height=0)
chosen1 = st.selectbox("Menu 1", ["Option1.1", "Option1.2", "Option1.3"], key="option1")
color_selectbox(0, 'pink')
chosen2 = st.selectbox("Menu 2", ["Option2.1", "Option2.2", "Option2.3"], key="option2")
color_selectbox(1, 'yellow')