Hi DariuszKominek-8002,
Cookie option is used in previous versions of SharePoint. This option is not available in newer SP. The users need to select their preferred language in their User Profile Settings.
Although the user profile settings could be changed by code, Unfortunately the updates won't immediately take effect in the UI, it may cost some minutes.
The following threads discussed the similar issue, you can take a reference:
- https://github.com/SharePoint/sp-dev-docs/issues/1636
- https://social.msdn.microsoft.com/Forums/office/en-US/d13dc319-b2e0-4a38-be02-6a36d4d2c0d3/language-switcher-in-sharepoint-online?forum=sharepointdevelopment
Best Regards,
Baker Kong