Yes, that's the proof, Anindya! Isn't it cute? You can see why identity elements are important: it doesn't work for mere semigroups.
I don't think there's a generalization to categories in the direction you're suggesting. The really nice generalizations to categories work a bit differently. You can read about them in my paper [Higher-dimensional algebra and topological quantum field theory](https://arxiv.org/pdf/q-alg/9503002.pdf).
Even if you don't want to read the paper, look at the picture at the top of page 25, and then the picture below that! This reveals the topological underpinnings of the Eckmann-Hilton argument.
One of the first spinoffs of this viewpoint is:
* A monoidal category in the 2-category of monoidal categories is a braided monoidal category.
* A monoidal category in the 2-category of braided monoidal categories is a symmetric monoidal category.
See, it goes on for one more step when we categorify it! This eventually leads to the "periodic table of \\(n\\)-categories", which is one of the main subjects of my paper.