It is very common to register a trademark that is a combination of a distinctive name, word or phrase and a logo or other design for one filing fee. This raises the question of whether the design or the name/word may also be protected as "registered trademarks" when used separately, and the answer is no. A registered trademark is "considered as a whole" and if you want to protect parts of it, you would be required to apply separately for registration of those parts.
A registration of a "standard character" mark may give much broader protection for a distinctive name than if it is only registered in combination with something else. Contrarily, a valuable (recognizable) logo may be the most important element, regardless of whether the name is with it.
In the USA, registration of a trademark is completely optional for enforcement in state or federal courts, although registration gives additional valuable rights, including "national preemption".
So, in theory, one could register a combination word-mark and design and then enforce the design (alone) or the word-mark (alone) outside of the registered rights (e.g., national preemption), and even place (tm) on it, but not ® (circle R), if it is not federally registered as shown.
There are numerous gray areas in most aspects of trademark rules for registration, let alone enforcement, often tied to subjective evaluation of "distinctiveness" or "related fields" that may "create a likelihood of confusion" among reasonable consumers in that market.