Работает ли это так, как вы хотели?
angular.module('myApp', []).controller('namesCtrl', function($scope) {
$scope.day_month_select_menu = [
{
id : 1, selectText:"day", show:"Monday", img:"tmp"
},
{
id : 2, selectText:"date", show:"Dec 12th 2018", img:"tmp2"
},
{
id : 3, selectText:"Test", show:"Dec 13th 2018", img:"tmp3"
}
];
});
<script src="https://cdnjs.cloudflare.com/ajax/libs/angular.js/1.7.5/angular.min.js"></script>
<div ng-app="myApp" ng-controller="namesCtrl" >
<select class="form-control" ng-model="selectedMenuOption" >
<option ng-repeat="menuOption in day_month_select_menu"
value="{{menuOption.id}}" >{{menuOption.id}} - {{menuOption.selectText}}</option>
</select>
<!--
<br/>selected:{{selectedMenuOption}}
<hr/>
-->
<ng-container ng-repeat="data in day_month_select_menu" >
<span ng-if="data.id == selectedMenuOption">
<br/>{{data.id}} & {{selectedMenuOption}}: <img src="{{data.img}}" alt="Error" />
<span>
</ng-container>
<ul>
<li ng-repeat="data in day_month_select_menu | filter:data.id">
{{ data.show }}
</li>
</ul>
</div>