selectedOptions: string[] = [];
onSelect(option: string) {
  if (!this.selectedOptions.includes(option)) {
    this.selectedOptions.push(option);
  }
}

removeOption(option: string) {
  const index = this.selectedOptions.indexOf(option);
  if (index > -1) {
    this.selectedOptions.splice(index, 1);
  }
}
<select (change)="onSelect($event.target.value)">
  <option value="" disabled selected>Select an option</option>
  <option value="Option 1">Option 1</option>
  <option value="Option 2">Option 2</option>
  <option value="Option 3">Option 3</option>
</select>

<ul>
  <li *ngFor="let option of selectedOptions">
    {{ option }}
    <button (click)="removeOption(option)">Remove</button>
  </li>
</ul>