The brand identity online it’s not just about a specific design, or a set of rules. It’s about the perception, the feelings that the user has and remembers throughout her or his experience with our website.
This perception is highly influenced by the performance (speed) and accessibility (e.g. responsive) of a resource.
A logo, banner or typeface can help and boost this experience, only if they’re not an obstacle between the information and the user, with their own (different) device.
Do not sacrifice accessibility in the name of brand identity.
Do not, ever, sacrifice accessibility. Websites are communication tools, they are meant to access information and to perform tasks online.