It may be a bit unhealthy to only identify yourself only with your vocation - there's also family, friends and hobbies. But it's miserable to engage in "work" for forty hours a week when it's something you have no interest in whatsoever. And it is very unhealthy to think most people should do that. If "work is just work", how terrible is that?

If the sum-total of the activities you engage isn't "you", what is? The things you buy? "Your" sports team?

Plus a hypercompetitive academic world is a pretty good explanation for people identifying only with their vocation - how else to do you get to the top except putting your whole self into the challenge?