Most of the communities in Stackexchange have dedicated logos.
But why don't we have one?
To the contrary, most of SE sites do not have a designed logo, as you can see by browsing the site list.
The post Graduation, site closure, and a clearer outlook on the health of SE sites outlines the process by which a site may acquire a custom design.
The site consistently gets at least 10 questions per day.
The Community Team considers some less quantifiable factors in deciding whether the site is ready to graduate; in particular, there should be enough engaged users to support a meaningful competitive election for 3 moderator positions.
After the site graduates, it joins the queue of sites waiting for a custom design. There are currently 10 sites in the queue.
Items 2-3 only become relevant after step 1. Amateur Radio currently gets about 2 questions per day.