For trademarks, the question of the ownership of the rights under an employment contract is less relevant. The person who registers the trademark - and this is generally a business - is the owner of the trademark rights. There are no specific legal provisions.
However, it is possible that the sign (e.g. logo, slogan or label) used as a trademark is original and designed by an employee or official or on request. In this event, the sign will also be protected by copyright and copyright regulations will apply.