Question: Forking on Github or Bitcket creates a repo's copy under your own account. 

Does this mean that forking is a server-side action? I was unable to find "fork command" inside Git.

If it's not a server-side action, what are the steps to fork a local Git repository with full history and tags?

Answer: 
*Forking* is not a technical operation but a process which happens in software development:

> 
> [...] a project fork happens when developers take a copy of source code from one software package and start independent development on it, creating a distinct and separate piece of software.
> 
> 
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*source: <http://en.wikipedia.org/wiki/Fork_%28software_development%29>*

Therefore, strictly speaking there isn't the notion of *forking* in git. That said, in order to fork a project that's in a Git repository, you have to `git-clone` it.
Forking is just copying the whole repo. `git clone` is how you do it locally. Or you could literally just copy the entire repo.

See also [Are git forks actually git clones?](https://stackoverflow.com/questions/6286571/git-fork-is-git-clone)