No you can't copyright fursonas. Despite what most people think, it is not possible to copyright your fursona.
The only thing copyrighted when it comes to fursonas is their art. Eg you receive a picture of it, the artist will always keep the copyright.
(I'm having this talk with people every few months)
WHAT you can do however, is register it as a trademark. That's going to cost you a bunch of money and needs to be renewed every few years. If I wanted to put a trademark on my fursona for Germany ONLY, it would cost me around 375$/7years. If I want the trademark to cover more countries, it's obviously going to be more expensive.
Also you need to remember that if you draw another person's fursona (or make a Minecraft skin out of it and IF it is trademarked), the laws in the creator's country apply, so they may be totally different than yours.
While you theoretically have the intellectual property when it comes to your fursona, proving it as yours can seem quite difficult, wherein the issue with copyrighting lies. Imagine all the Tigers with white spots on black ears as only defining marking trying to sue each other (to bring an example ridiculous enough to make a point)
ENTER FURRY FANDOM
In the furry fandom people are usually nice enough to each other to respect individually created fursonas. People stealing others' designs are mostly frowned upon and quickly loose hold in the fandom. Most tigers with white spots on their black ears just accept it as a widely used (naturally occuring) marking, just as huskies accept most of them are grey&white.