How do I capture the selected option and pass in sending the registration form?

Way use jQery to populate the select field when you can use filter and hook into the form using PHP eg:

 //1. Add a new form element...
add_action('register_form','_register_form_wpa103118');
function _register_form_wpa103118 (){
    $name_of_select = ( isset( $_POST['name_of_select'] ) ) ? $_POST['name_of_select']: '';
    $options = array(
        'Medicine' => 'Medicine',
        'Option2' => 'Option2',
    );
    ?>
    <p>
        <label for="name_of_select"><?php _e('First Name','mydomain') ?><br />
            <select name="name_of_select">
                <?php
                foreach ($options as $key => $value) {
                    $selected = ($value == $name_of_select)? ' selected="selected"': '';
                    echo '<option'.$selected.' value="'.$key.'">'.$value.'</option>';
                }
                ?>
            </select>
    </p>
    <?php
}

//2. Add validation. In this case, we make sure first_name is required.
add_filter('registration_errors', '_registration_errors_wpa103118', 10, 3);
function _registration_errors_wpa103118 ($errors, $sanitized_user_login, $user_email) {
    if ( empty( $_POST['name_of_select'] ) )
        $errors->add( 'name_of_select', __('<strong>ERROR</strong>: You must select ....') );

    return $errors;
}

//3. Finally, save our extra registration user meta. or do whatever you want with it.
add_action('user_register', '_user_register_wpa103118');
function _user_register_wpa103118 ($user_id) {
    if ( isset( $_POST['name_of_select'] ) )
        update_user_meta($user_id, 'name_of_select', $_POST['name_of_select']);
}