No, trademark classes categorize goods and services the brand offers. You can learn more about them here.
When registering both the brand name and the logo, you have two options:
- You can file two separate trademark applications, or
- if your logo includes the name and the name can be considered sufficiently distinctive and not merely descriptive of your products/services, you can choose to register just your logo, since the name will inherit some level of protection as well
In either case, any of these trademark applications would have to be filed in the same classes. For example, if you have a clothing brand and want to register both the name and the logo separately, you will effectively apply for two trademarks - one figurative trademark in the Class 25 (Clothing), and one word mark in the same Class 25.