You must define validateSid handler to make use_strict_mode work.

It doesn't work either.

I can't use
class MySessionHandler extends SessionHandler
{
    public function validateId($key){
        return parent::validateId($key);
    }
}
because SessionHandler::validateId() is not defined.

Is there any method to wrap internal 'files' handler with use_strict_mode?

Because of ignoring use_strict_mode flag when 'files' session handler is exposed, custom validateId method must be implemented that depends on internal realization of file storage mechanism.

This is a minimal implementation:
<?php
session_module_name('files');
ini_set('session.use_strict_mode', '1');

session_set_save_handler(new MySessionHandler(), true);

session_start();
var_dump(session_id());

class MySessionHandler extends SessionHandler
{
    /**
     * Getting full path to session file
     *
     * It's depend on internal realization of 'files' hander.
     * Must be reviewed after each php release.
     */
    private function getFullPathName(string $key) : string
    {
        if (empty($filename = session_save_path())) {
            $filename = sys_get_temp_dir();
        } elseif (strpos($filename, ';') !== false) {
            $data = explode(';', $filename);
            $count = $data[0];
            $filename = end($data);
            for ($i = 0; $i < $count; $i++) {
                $filename .= '/' . $key[$i];
            }
        }
        return $filename.'/sess_'.$key;
    }
    
    public function validateId($key)
    {
        return file_exists($this->getFullPathName($key));
    }
}

I've run into this issue myself and believe it is a bug. The SessionHandler class is described as "a special class that can be used to expose the current internal PHP session save handler". So the expectation is that using SessionHandler as-is will provide the same functionality as if you had not set any save handler, but in practice this will disable the 'use_strict_mode' option.

session_module_name('files');
ini_set('session.use_strict_mode', '1');
session_set_save_handler(new SessionHandler(), true); // This breaks strict_mode

The root issue is that the SessionHandler class does not implement validateId().
With no other way to expose the internal handler for validating the id, we are forced to figure out how the internal files storage works and implement the validation manually.