Because he was the leader (not yet Emperor) of a republic founded on the ideals of liberty, equality, and fraternity, and answering egalitarian objections to his creation of the Legion of Honour, not writing a treatise on forms and principles of government.

I suspect that it helped to signal his status as a member of the philosophical tradition of the Enlightenment, and of the French Revolution that was framed in Enlightenment terms. To the extent that the Enlightenment was focused on experimentation with new forms of government, its adherents tended to favor the republic structurally. (Not universally, of course). He also pushed through a number of reforms with the same flavor, and his status in that group seems to have accounted for at least some of his popularity in France.

There may also be a subtle religious dimension here. Recall that the Enlightenment entertained Deism and a comparatively radical heterodoxy; using the word 'republic' might have evoked pagan Rome, great icons of the Renaissance (i.e. Florence), and his own enlightened France, while neatly skipping over the most notable Christian theocratic/monarchist governments.

Probably because he was responding to people who were arguing that these things were inappropriate in a republic.