But other branches of engineering are also about doing a lot of repetitive work, exactly what "programmers" do.

Engineers shouldn't compromise on things like safety, security, legislation, budget, properly documenting, properly communicating with others, etc. Same as anyone working professionally with code should, no matter the title.

But there's nothing that says that engineers should only be doing groundbreaking or interesting work.

The "artist" analogy works much better.

EDIT: Perhaps the main difference is what psychoslave mentions below – engineers are expected to question the relevance and necessity of requirements, and work together with business, rather than just doing as asked.