Try this

Define a state (bgColor) inside the parent component (Landing) and pass controllers (setBgColor) to the children (buttons).

import { useState } from "react";

const Landing = ( { children }) => {

    const [ bgColor, setBgColor ] = useState("blue");

    return (
        <div className={container} style={{ backgroundColor: bgColor }}>
            <div className={wrapper}>
                <ZoneButton zone={'pink'} icon={'child'} setBgColor={setBgColor}/>
                <ZoneButton zone={'blue'} icon={'code'} setBgColor={setBgColor}/>
            </div>
        </div>
    )
}

set the bgColor depending on the zone

const ZoneButton = ({ zone, icon }) => {

    const colourPrimaryDict = {
        'pink': 'var(--pastel-pink-primary)',
        'blue': 'var(--pastel-blue-primary)'

    }
    const colourSecondaryDict = {
        'pink': 'var(--pastel-pink-secondary)',
        'blue': 'var(--pastel-blue-secondary)'
    }
    const svgDict = {
        'code': <svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="code" className="svg-inline--fa fa-code fa-w-20" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 640 512"><path fill="currentColor" d="M278.9 511.5l-61-17.7c-6.4-1.8-10-8.5-8.2-14.9L346.2 8.7c1.8-6.4 8.5-10 14.9-8.2l61 17.7c6.4 1.8 10 8.5 8.2 14.9L293.8 503.3c-1.9 6.4-8.5 10.1-14.9 8.2zm-114-112.2l43.5-46.4c4.6-4.9 4.3-12.7-.8-17.2L117 256l90.6-79.7c5.1-4.5 5.5-12.3.8-17.2l-43.5-46.4c-4.5-4.8-12.1-5.1-17-.5L3.8 247.2c-5.1 4.7-5.1 12.8 0 17.5l144.1 135.1c4.9 4.6 12.5 4.4 17-.5zm327.2.6l144.1-135.1c5.1-4.7 5.1-12.8 0-17.5L492.1 112.1c-4.8-4.5-12.4-4.3-17 .5L431.6 159c-4.6 4.9-4.3 12.7.8 17.2L523 256l-90.6 79.7c-5.1 4.5-5.5 12.3-.8 17.2l43.5 46.4c4.5 4.9 12.1 5.1 17 .6z"/></svg>,
        'child': <svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="child" className="svg-inline--fa fa-child fa-w-12" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 384 512"><path fill="currentColor" d="M120 72c0-39.765 32.235-72 72-72s72 32.235 72 72c0 39.764-32.235 72-72 72s-72-32.236-72-72zm254.627 1.373c-12.496-12.497-32.758-12.497-45.254 0L242.745 160H141.254L54.627 73.373c-12.496-12.497-32.758-12.497-45.254 0-12.497 12.497-12.497 32.758 0 45.255L104 213.254V480c0 17.673 14.327 32 32 32h16c17.673 0 32-14.327 32-32V368h16v112c0 17.673 14.327 32 32 32h16c17.673 0 32-14.327 32-32V213.254l94.627-94.627c12.497-12.497 12.497-32.757 0-45.254z"/></svg>
    }

    return (
        <div className={container} style={{
            backgroundColor: colourPrimaryDict[zone],
            color: colourSecondaryDict[zone]
        }} onClick={() => setBgColor(zone)}>
            {svgDict[icon]}
        </div>
    )
}

You can use this to change the className if you need.

You can do something like this:

const ZoneButton = ({ zone, icon }) => {

    let [colourPrimaryDict, setColourPrimaryDict] = React.useState({
        'pink': 'var(--pastel-pink-primary)',
        'blue': 'var(--pastel-blue-primary)'
    })

    let [colourSecondaryDict, setColourSecondaryDict] = React.useState({
        'pink': 'var(--pastel-pink-secondary)',
        'blue': 'var(--pastel-blue-secondary)'
    }) 
    const svgDict = {
        'code': <svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="code" className="svg-inline--fa fa-code fa-w-20" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 640 512"><path fill="currentColor" d="M278.9 511.5l-61-17.7c-6.4-1.8-10-8.5-8.2-14.9L346.2 8.7c1.8-6.4 8.5-10 14.9-8.2l61 17.7c6.4 1.8 10 8.5 8.2 14.9L293.8 503.3c-1.9 6.4-8.5 10.1-14.9 8.2zm-114-112.2l43.5-46.4c4.6-4.9 4.3-12.7-.8-17.2L117 256l90.6-79.7c5.1-4.5 5.5-12.3.8-17.2l-43.5-46.4c-4.5-4.8-12.1-5.1-17-.5L3.8 247.2c-5.1 4.7-5.1 12.8 0 17.5l144.1 135.1c4.9 4.6 12.5 4.4 17-.5zm327.2.6l144.1-135.1c5.1-4.7 5.1-12.8 0-17.5L492.1 112.1c-4.8-4.5-12.4-4.3-17 .5L431.6 159c-4.6 4.9-4.3 12.7.8 17.2L523 256l-90.6 79.7c-5.1 4.5-5.5 12.3-.8 17.2l43.5 46.4c4.5 4.9 12.1 5.1 17 .6z"/></svg>,
        'child': <svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="child" className="svg-inline--fa fa-child fa-w-12" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 384 512"><path fill="currentColor" d="M120 72c0-39.765 32.235-72 72-72s72 32.235 72 72c0 39.764-32.235 72-72 72s-72-32.236-72-72zm254.627 1.373c-12.496-12.497-32.758-12.497-45.254 0L242.745 160H141.254L54.627 73.373c-12.496-12.497-32.758-12.497-45.254 0-12.497 12.497-12.497 32.758 0 45.255L104 213.254V480c0 17.673 14.327 32 32 32h16c17.673 0 32-14.327 32-32V368h16v112c0 17.673 14.327 32 32 32h16c17.673 0 32-14.327 32-32V213.254l94.627-94.627c12.497-12.497 12.497-32.757 0-45.254z"/></svg>
    }

    let changeColor = () => {
        //Change your color here
        setColourPrimaryDict({
            'pink': 'var(--pastel-pink-primary)',
            'blue': 'var(--pastel-blue-primary)'
        })
    }
    return (
        <div className={container} style={{
            backgroundColor: colourPrimaryDict[zone],
            color: colourSecondaryDict[zone]
        }} onClick={ () => changeColor() }>
            {svgDict[icon]}
        </div>
    )
}

On click just change the state value.

It is so simple. You just understand what is happening in react for bypassing the props in components. This is a small snippet for better understanding.

Change the passed props state value, which is sent by the parent component or use the redux for global declaration

Parent component

function App() { 
  const [data, setData] = useState("#fff")
       return (
      <div style={{
           backgroundColor: data
      }}>
         <Test  setData={setData}/>
      </div>)
}

Child component

function Test(passedProps) {
   let {setData}  = passedProps
   return (<button onClick={() => setData("#f1f1f1") }>click to change color</button>);
}