The theory of relativity, as we understand it today, would preclude faster-than-light travel altogether, and make near-light travel subject to time dilation, as you said, which makes for some very good science fiction, but really gets in the way of space opera, of fast ships travelling to and fro around the galaxy.
To achieve that, various science fiction authors and franchises have looked for ways to bypass relativity and its pesky limitations. These ways usually take the form of some sort of warp drive or hyperdrive or hyperspace travel, which basically means that the limits of relativity remain - in our dimension. But the new drive allows your ship to move to a different dimension/plane/hyperspace, travel a much shorter distance, and return. This leads to various effects like "rifts" between that alternate dimension and the regular one.
So yes, you can say that Einstein's theories are ignored - or at least, explained away with some technobabble and handwaving - for the sake of the story. The consistency, coherence and style of that handwaving is what makes for a believable sci-fi drive.