Sauce Labs actually supports a REST API for setting job names; You can over-write the name you set with your desired capabilities using that REST API.
A RemoteWebDriver, however, works exactly as you said: Each new WebDriver opens a new browser. Just like with a local browser, each new RemoteWebDriver connection to Sauce Labs creates a new VM. If you run all your tests against one VM, they'll all end up in the same Sauce Labs job. Each job can only have one name.
Probably the best way to test efficiently with Sauce Labs is to run multiple tests in parallel; If you really want to use the same job for all your tests in turn, you could use the "sauce:context" javascript method to give each section of that job a name, eg:
IJavaScriptExecutor js = driver as IJavaScriptExecutor;
js.ExecuteScript("sauce:context='Test The Friend Page Works'");