I have a little problem because my fursona is named the same like my aritst's name (btw. My username is not that name) and normally I would tag the works including my fursona this way:
"example_(artist)" and "example_(character)"
But...
There is the guy who's fursona is named the same like mine.
What should I do? Can't tag my character like "example_(example)" because it looks just weird.

Perhaps you could think of a last name to make the tag unique to your character? So instead of being example_(character), or in your case example_(example), the tag for your character would be example_lastname.

Yes, it would work but If anyone would want to search my character by tag, probably will find other guy's character first. Can I change then tag of this guy from "fursonaname" to "fursonaname_(artistname)"? Then the tag would be "empty" like this one - https://e621.net/post/index/1/john

There's nothing wrong with name_(name) :)

Especially if there's ANY likelihood that there might be another character who shares that name. Seriously, no one, NO ONE wants to be the one to 'fix' john_(character)

I've been there, I've done that and it's not any fun. It involves a lot of looking at sources and trying to figure out who owns THIS version of shadowfox, and even 40 or so pictures takes an hour or two to get sorted.

So, while SnowWolf_(snowwolf) might look weird, it's specific and to the point and it's incredibly unlikely that you'll have two snowwolfs (...snowwolves?) which have characters named Snowwolf as well.

Unlikely, anyway. Then again, when I was a wee thing, my fursona and username were basically the same. I went by Wolf. And when THAT was taken, I took up "GrayWolf"... Oddly, it's pretty hard to find any evidence of me on the early furry internet. Can't imagine why. Thankfully, I'm much more creative these days. ;)

So, yeah: be clear, don't worry about things looking weird.

edit: also, be fair: if your'e gonna name_(owner) one person, you name_(owner) all of them :) Unless there's a LOT more of one characters art than another. There are a lot of characters that will ever only exist in 2-3 pictures.