[Vuejs]-Highcharts – How to set Pie Chart's background color dynamically

You should do an update on the whole chart config, not on the series:

chart.update({
    chart: {
        backgroundColor: 'red'
    }
})

API: https://api.highcharts.com/highcharts/chart.backgroundColor

API: https://api.highcharts.com/class-reference/Highcharts.Chart#update

Demo: https://jsfiddle.net/BlackLabel/p7yg386x/