A brand is when the name of a product, service, concept, etc. has been filed with the INPI and when its holder may be opposed its use by a third party in a given territory (France, Europe or the whole world).
The trade name or legal name is the legal term for the name of a company.
You can use a trademark as a name for your company but it is not mandatory.
Choose the name you like (for your products or services and/or for your company) and think of 3 important things:
- Use the letters of the Latin alphabet, without any special characters
- Check that the name is available by doing a little research on the site of the INPI
- Check that it is not prohibited
To choose well, this article may help you