It's a matter of the so called "Nizza class" - there are a lot of different "classes" for different types of things, like "drinks", "electronic gadgets" and so on. Each class has a number assigned to it.
It is possible to register "MySuperBrand" as "drink", while another company already has registered "MySuperBrand" as "electronic gadgets".
So even if the brand is already registered, it might be for another class, so you are free to register it for your needs. Be aware that large brands, even if they did not register all possible classes (as they have to show active use of the class), might be unhappy if you use the same brand name, which usually involves lawyers. So it's best to ask a trademark lawyer in advance. Pro tip: it's not clever to register "Apple" as a brand for armchairs, even though Apple does not sell armchairs.
You can trademark a "name", which is much more restrictive than registering "name and picture" or "picture". So if you register "Coca-Cola" as a name, it's irrelevant which font you use. If you register the nike swoosh as a picture, only that exact swoosh is protected. Make it longer, thicker, whatever - in theory the trademark will not cover it.
But: all of this is not exact science, it's always subject to interpretation - and money (lots of money) if you need to defend yourself. Again, ask a lawyer.
Edit: If the website is already registered, as a brand owner you have higher rights to it than a non-brand-owner. Enforcing might not be easy, depending on the domain.