YThe question isn't if you need to run a setting strictly or if you need to write up your own stuff. Not at all. It's a question of the definition of a setting - if you used Greyhawk maps and some Eberron history and culture, mapping the last war and the mournland and some of the rest onto Greyhawk, are you running Greyhawk? My guess is most people would say no.
So the lore of a Eberron warforged are <snip lore details>
Take out all of the lore and what do you have? A set of rules for a race. You do not have an Eberron warforged. You can replace their lore just fine - it's all allowed. A camapign I'm currently running calls them the Dwarfforged and has a very different creation story for them, and they are their own unique thing. But no one would mistake it for an Eberron Warforged, because they are not.
It seems to me that when you say "most people would say no", or that "no one would mistake it for an Eberron Warforged", you may not be primarily making empirical conjectures about how RPGers would use labels.
It seems that maybe you have a view about what it is for something to be a GH game, or an Eberron warforged, and you think that what is described doesn't fit you conception of these things.
Which is fine, but I'd be interested to hear you elaborate a bit more what your conception is. And also what, exactly, is at stake.
To elaborate on what I mean by "something being at stake" - suppose we debate whether cubism should be seen as a form of post-impressionism, or rather as a type of anti-impressionism. What's at stake? Well, we're trying to make sense (intellectual sense, historical sense, aesthetic sense) of trends in modern, post-the-invention-of-photography visual arts.
Or for a completely different context: when the staff at WotC debate whether or not some module or adventure or supplement should be released under the FR label, what's at stake? For them, it's a commercial issue - how can they (i) monetise the FR "brand", without (ii) diluting that brand so that monetisation becomes harder. It's a type of optimisation problem which turns on the fact that publishing stuff under the FR label simultaneously exploits and shapes certain consumer preferences.
But when we ask whether or not my GH game - which I freely describe as a GH game, both to my friends and online - is "really" a GH game, what is at stake? What question are we trying to answer, or problem are we trying to solve, by reaching the view that it is or isn't? (And likewise for the warforged.)
For what it's worth, when I tell my players we're playing in GH, and then one of them who has no prior familiarity with GH Googles some maps; and fills in the Ancient Languages slot on his PC sheet as "Ancient Suel" because Google told him that, in GH, the Suel are the ancient magical cutlure; I count that as a labelling success. Labelling it a GH game provided some effective short cuts to establishing some elements of the shared fiction.