Stack Exchange sites go through a number of stages. First, they are defined through example questions, and users commit to participate in building the site. This happens on Area 51. Then the site is actually created, spends a few days in private beta and then a few months to a few years in public beta.
During the beta phase, the site is “on probation”; all Stack Exchange sites in beta have the same visual appearance.
When the site reaches a certain threshold of traffic, it “graduates”, indicating that the site is here to stay and isn't likely to undergo major changes in scope and culture. We graduated last November. The final stage after graduation is getting a distinctive design; for a long time it used to happen at the same time as graduation, but these days it's a gradual process.
As it happens, we are about to get a distinctive look.
Stack Exchange sites don't normally have a distinctive name, just “<Topic> Stack Exchange”. Stack Overflow came first (Stack Exchange got its name as a generalization of Stack Overflow); then came Server Fault and Super User; then the “Stack Exchange 2.0” process which allowed other sites to be created. It was originally planned that each site would get a fancy name, but quickly this decision was overturned. So we're getting a logo, but we'll keep being called “Emacs Stack Exchange”.