Recently I was asked that very question regarding how I thought of myself as a front end engineer. Do I have to be either or? Is it possible to simply be a front end engineer without having to be a transplant from another discipline?

I think this depends greatly on where you sit. Coding is very helpful to me as a designer, but the primary value I add is in design. Coding is a "nice to have" for me, and most see it that way. In fact, I've had employers disuade me from coding so I'll keep my focus on design. I don't like that very much as I enjoy learning things and coding helps me bring my designs to life.

I'm not sure it makes as much sense for an engineer to have "design sense" (whatever the heck that means.)

Being able to talk to designers helps. Having understanding of how engineering effects UX helps. A product-minded approach helps.

If you can do both, great, but I do think it makes sense to "pick" one or the other to focus on.

And, yes, I think it's possible.

In some circumstances I could see this as a question to determine strengths; answer "designer" it means you're not code is weak, "engineer" and you should stay away from Photoshop. If that's the case, it would make more sense to simply ask what my strengths are.

I'm in total agreement with what you're saying about designers who understand code and engineers with some design skills/sensibilities; or at least an ability to communicate with each other effectively.

I definitely don't think you have to be one or the other, but the biggest, I think, benefit of having aspects of either (both?) is speaking the language.

Understanding both groups use similar (sometimes the same) words for wildly different things means you can translate conversations, requirements, and feedback much more efficiently and (one hopes) without antagonism.