TYPO3 Core

ext_localconf.php

<?php
defined('TYPO3_MODE') || die();

$GLOBALS['TYPO3_CONF_VARS']['EXTCONF']['ext']['fields'] = [
    'op1' => 'Option 1',
    'op2' => 'Option 2',
];

ext_conf_template.txt

# cat=basic; type=user[Vendor\Ext\Utility\ExtensionManagerConfigurationUtility->buildFieldsSelector]; label=Field
field =

Vendor\Ext\Utility\ExtensionManagerConfigurationUtility

<?php
namespace Vendor\Ext\Utility;

class ExtensionManagerConfigurationUtility{
    /**
     * @param array $params
     *
     * @return string
     */
    public function buildFieldsSelector(array $params): string {
        $selector = '<select id="em-'.$params['propertyName'].'" class="form-control" name="'.$params['fieldName'].'">';
        $selector .= '<option value="">'.$GLOBALS['LANG']->sL('None').'</option>';

        if (count($GLOBALS['TYPO3_CONF_VARS']['EXTCONF']['ext']['fields'])) {
            foreach ($GLOBALS['TYPO3_CONF_VARS']['EXTCONF']['ext']['fields'] as $v => $k) {
                $selectd = $params['fieldValue'] === $v ? ' selected' : '';

                $selector .= '<option value="'.htmlspecialchars($v).'"'.$selectd.'>'.$GLOBALS['LANG']->sL($k).'</option>';
            }
        }

        $selector .= '</select>';

        return $selector;
    }
}

When you go to BE > Settings > Extension Configuration > Configure extensions, you will get the Ajax error.

If the $GLOBALS['TYPO3_CONF_VARS']['EXTCONF']['ext']['fields'] is moved to LocalConfiguration.php, the Configure extensions will work as expected. So, my guess is ext_localconf.php is not loaded or what.