If I created an icon for a site on SE, how could it become the official one? (Instead of the light-blue speech bubble with initials in it)
You can't. There is a dedicated design team in the company that creates the brands and logos. You can't just add your own. You can propose a change though through the per-site meta.
If you don't like this one, feel free to propose a change. Not likely it is implemented though.
On beta sites, there are standard logos used. Eventually, somewhere after a site has graduated, the design is made and implemented.