I don't care about Dual Spec one way or the other, but this is somewhat flawed logic with Vanilla's class balancing. It isn't really "I'm an Arms Warrior or Fire Mage" so much as it's "I'm a mage (and that means I play frost because fire is non-viable in early content, or fire in later tiers)" or "I am a warrior (and that means prot, or arms without gear and fury with it" or "I am a druid (and that means resto)".
This "I am a <spec> <class>" thing only really showed up when you were prefacing the fact that you were playing something less than great, like ret or balance, because then people are likely to be angry if you don't point out that you are playing an unusual spec. It's not like when someone was LFDPS you whispered them "Hey, demonology warlock here." either the content was low enough that spec was irrelevant, or you were playing the best spec because you weren't a dummy.
I'm not sure dual spec really does all that much to alter identity, especially when for a lot of classes, dual spec would mean having a raiding talent setup for your best spec, and having a PvP talent setup for your best spec. I don't think any priest who dual specs holy-shadow, and heals in raids (lol1shadowpriestonlymaybe) but uses shadow for farming and low content is going to have an identity crisis. They are a priest (holy). That's what they raid as, and do relevant content as.
Particularly with the modern day, "Anything but the most optimal is trash" mindset, I don't expect you'll see all that many people respecing often to begin with, unless they are hybrids who are also PvPing/farming often in the open world, in which case they are going to respec back and forth anyway, dual is just making their life less of a pain in the ass.
But again, I don't really care either way. I didn't switch specs often during Vanilla.