How to control the select box when I add new one?

You need to put $('.selectpicker').selectpicker(); as last line of addColorOptions method to initialize the newly added select.

As we are adding a bootstrap-select dynamically, we need to initialize them after creating them. You can find the documentation bootstrap-select here

when everything is working fine and just dynamically added select inputs not rendering as expected

 function addColorOption() {
    var color = `
        <div class="form-group">
            <select class="selectpicker" data-live-search="true" name="color_id" style="width:100%">
                @foreach($colors as $color)
                    <option value="{{ $color->id }}">{{ $color->name }}</option>
                @endforeach 
            </select>
        </div>
    `;
    $('#products-color').append(color);
    $('.selectpicker').selectpicker();  //just adding it here will do
}
``