I think what the OP means is the fear of losing a "panmage" identity in favour of other unifying identities.
As you say, mage identity exists even in non-Circle societies and it will exist as long as there are people with magic and people without it. However, we also know that in several of those cases that mage identity takes a second seat to another union-driving identity. It's not very different in real life with nations, religions, political ideologies, etc.
The Circles, by their own existence, provide not only a clear, defined mage identity, but the only instance so far of a broader panmage identity, that is, an identity that puts "I'm mage" first and it's expected to apply universally to all of them. The end of the Circle could well mean the end of that.
Not necessarily- or rather, not unless another identity exists to supplant it. Identity politics, like inertia, continue until something else impacts it with more momentum. There's nothing in particular waiting to do that at this time.
At this point, the only example we have of a mage class in which being a mage wasn't part of being the primary identity is the Dalish. It's not clear what level of identity exists in Rivain, but we may also have male mages there as well. But both of these are contexts which developed naturally over tens or even hundreds of hundreds of years. That obviously will not apply to the Circle mages in the next five, ten, or even fifty.
That would be what already happens, but with the Templars becoming clever administrators instead of just wards. But I doubt that it could erase the panmage identity.
Eh, not really. Or rather, that's not what already happens. The Fraternities exist as political sub-factions, but not as independent polities, and the Templars and Chantry have never, to our knowledge, played them against eachother for the purpose of anti-mage identity. Anti-mage unity is different, inherent with the fact that these are non-unified political positions, but unity and identity are enormously important distinctions.
Playing factions against identity goes beyond clever administration and into outright police state controls. Think the Soviet Union's destruction of various minority identity groups. (That may be hard, of course, since the nature of cultural erasure is that it's the failures that are still around to be called to mind.)
For that to happen, they shouldn't be treated equally from the start, but how can you expect to do that when you have to take children "because they are mages", not "because they will become part of a fraternity in the future"? Calling it "Circle mage" or "Fraternity mage" is irrelevant, because the main distinction will always be "mage".
Easily, if you're willing to be ruthless enough. You make the differences more compelling and relevant than the universal commonality. Everyone of them is a mage... but then, mages and mundanes are all people too, and the pan-identity can and was still broken.
Think of it as a reverse case of Doctor Sues's The Sneetches. Create a new class/caste/identity system, and treat it as if it's very important, and in short order the members of the arbitrary groups will as well. Only in this case instead of an invented categorization, use real categorical differences, and play them up.