Moving sideways is allowed in fencing. The modern piste is typically 1.5 - 2m wide. It doesn't sound like a lot, but fencing is very fast, and when someone is attacking you, you honestly don't have time to move more than a few inches to one side or the other, so it's actually plenty. Here is an example of what a side-step looks like in the modern game--and still he wasn't fast enough, and got touched.

At longer distances you have more time to move side-to-side, of course, but at longer distances you can't deliver a hit, and the opponent has a much smaller correction to make to track you, so all that side-stepping does not gain any advantage. The only real reason to do it is if you wish to make use of obstacles, terrain, or the direction of the sun to achieve more advantageous ground to fight from. However, in a formal duel of honour, which is where most thrusting swordplay was put to use, special care was taken to ensure a fair and equitable duelling ground, so that wasn't really an option.

The trend to linearity has a simple explanation: when there are only two fencers kept at a fair distance by the length of their weapons, all of the action occurs on the line that joins them. Therefore, one-vs-one fencing is intrinsically linear in its nature, especially so with thrusting weapons. If the fencers circle each other at safe distance, the line simply rotates to little tactical effect. The Italian masters of the early 17th Century realized this and formalized it, and the Italian systems of the time were recognized for their linearity. This is in contrast to the competing Spanish systems (destreza) which generally placed the combatants on a circle and encouraged sideways movements, even though their attacks were similarly linear chords across this circle. But it was the Italian systems that prevailed historically, and modern fencing owes a lot to those Italian roots.

Now, we could in principle fence on a wide-open floor and allow movement in any direction, but the tactics would always reduce down to fencing on that line. The line might rotate and swing about from time to time, but that doesn't change the fundamental geometric equation. And it would be a much less efficient use of space, which has been a concern of fencing masters trying to extract maximum value from their students for hundreds of years ;-)

Incidentally, you can see fencing's intrinsic linearity even in an open-floor system like kendo (here's an example)--little effort is wasted in superfluous circling movements, the fencers are quite linear in their approaches, except at the very moment of the strike, when the line of attack pivots suddenly, and the two fencers re-orient to it.

So all of these considerations have led fencing to formally restrict the playing field to the line. The piste itself has varied in length and width over the years, but those are minor technical variations. The price is that fencing must be stopped and restarted following actions where the fencers close too far and the line of attack pivots off the piste, but that is generally regarded as a minor inconvenience.

Lookup Narváez, Thibault, Destreza, and the Mysterious Circle, if you are curious about the history of circular fencing fencing systems. And for competing linear Italian systems based on the lunge: Agrippa, Capo Ferro, Giganti, Fabris, and many others.