Recently, we were asked how do we Remove/Update the "Account Name" displaying after an Agent's name.
Although there is no way (currently) to make it dynamic based on Brand (hello, Zendesk Multibrand PMs are you listening?), you can change this to something generic that does the job.
While logged in as a Zendesk Administrator, go to Settings -> Account -> Branding and update the name in "Your Zendesk account name". This will not affect any other part of your Support product.