如下即可，style需要套一层{}
XXXX就是typeMap对象里面的aaa或者bbb

<div class="custom-icon" :style="{ background: typeMap[XXXX]?.background }">        
</div>

const typeMap = {
  'aaa': {
    icon: 'icon-aaaa',
    background: 'rgba(111,111,111,0.11)',
    color: '#ffffff',
  },
  'bbbb': {
    icon: 'icon-bbbb',
    background: 'rgba(222,222,222,0.22)',
    color: '#aaaaaa',
  },
};