You haven't given any reason as to how this would work, why you suggest it, or why this would be useful/beneficial.
This counter is to count is for public people so anyone who doesn't
have SE account can upvote an answer he had a use of it.
And what happens to the vote from the user not logged in?
Does it get added to the current vote count, just like votes from users who are logged in?
Or are you suggesting a new additional vote count for "votes by non-logged in users"?
EDIT: I think SE can use Local Storage to store some id or something
so the answer doesn't get re-publicly-upvoted due to dynamic IP change
You are getting ahead of yourself a little bit.
You're trying to solve technical problems when you haven't even explained what this is all for.
Such potential problems can be ironed out by Stack devs, what you need to do at the moment is explain your request/proposal/ideas so we know what exactly you are proposing and why, and how it would work etc.
Currently, you idea is to "allow non-logged users a special vote button", but what does it do, and why does it do it?