Did folks say, "Nineteen hundred [and] eighteen?" I think not. So I'm curious why I consistently hear "Two thousand [and] eighteen." It's become enough of a peeve for me to consult you, Doc. C'mon, people, say it with me: "Twenty-eighteen."—Eats Shoots and Leaves
I'm of two minds about how to approach your question, Shoots. The language nerd in me would like nothing better than to pour myself a tall glass of bark dust and settle into a leisurely 42-hour exploration of English date-pronunciation conventions from 500 AD to the present.
However, the slightly better-socialized part of me suspects that, were I to succumb to such a temptation, I would resurface to find the rest of humanity had either fled the planet or torn out their own eyeballs from sheer boredom.
So I'm going to answer your painfully specific grammar-and-usage question, but in a way that keeps it, you know, sexy.
For starters, people absolutely said "nineteen hundred and eighteen." In fact, Nobel laureate W.B. Yeats has a very famous poem called "Nineteen Hundred and Nineteen," all written out to make sure you say it properly. We tend not to use this locution today, but then again, many ingenious lovely things are gone that seemed sheer miracle to the multitude.
There's an even more obvious reason why we say "two thousand eighteen." If we'd followed Shoots' "twenty-whatever" rule starting at the beginning of the century, 2005 (for example) would have been pronounced "twenty-five," which, let's be honest, would have been completely useless.
Thus, for the first 10 years of the century, we had no choice but to pronounce the year as "two-thousand (whatever)." (One hopes that the folks who insisted that "aught"—as in "twenty-aught-six"—was bound to catch on have finally made peace with the fact that it's not going to happen.)
Given this constraint, it's not surprising that the habit of beginning dates with "two thousand" has persisted into this decade. That said, you will note that the year 2020—which comes up a lot, being the year in which many hope to see the last of Donald Trump—is almost universally pronounced "twenty-twenty." So relax and drink your bark dust—it's gonna work out, eventually.