You're a little confused on the relationship of Fate to Fate Core. This is a bit like proposing we merge dungeons-and-dragons or d20 with dnd-5e.
The Fate RPG stretches back as far as 2003. It had an original standalone iteration, then was implemented as the foundation in many different games through the years like Spirit of the Century and Dresden Files RPG. Eventually in 2013 Evil Hat produced a definitive standalone "core" edition of the game, appropriately named Fate Core, which consolidated various mechanical improvements that'd been developed so far in its various iterations, and made some further improvements, simplifications, and clarifications of its own. This is just one version of the Fate RPG, you see, and it's now being implemented and improved on in subsequent games, so I expect by the year 2030 we'll see a Fate Core 2.
So, no, we shouldn't merge them.
As for how we're tagging these questions:
Questions tagged fate are asking about issues inherent to all Fate editions. (There are no “just Fate” rules, but that doesn't prevent there from being “just Fate” issues.) Questions tagged fate-core are asking about issues inherent to that specific edition.
These are two different scopes, and a question can reasonably be about just one or the other or both.
There might be some consistency we can work on in how we tag, but that's separate to the merge proposal presented here, and we can talk about that in a separate question.