Sorry if this is also tl;dr..!

The reason the game is playing with the "black bars" on either side of your screen is due to the resolution you chose not being mathematically friendly with the actual dimensions of your monitor screen/the resolution your computer itself is set at. It's for the same reason that when we watch non-HD channels on an "HD" tv (or 1920x1080) that those black bars are there as well (unless your tv or provider has a setting in which you can "stretch" the picture, of course...), considering the resolution that the video was originally produced to be in was intended for older, smaller, "square-er" TVs.

I had the same issue as you with my laptop, and so I set it to 1366x768. This is because it's a resolution that's lower and friendlier on the FPS issue, and because I can see the text easier on my TV, haha. However, the other reason is that it divides evenly from 1920x1080.

To be more specific... We're going to use an equation a lot of people learn to find out what their percentage was on their school tests: dividing their score by the goal score, heh.

Divide 1366 (the length of one of the FFXIV resolution choices, and the one I chose) by 1920 (the length of your resolution). You'll find the result is 0.71 rounded off. Do the same with the height of both resolutions: 768/1080. It also equals 0.71.

So! 1920x1080 is 29% larger than 1366x768, as 0.71 means that the smaller resolution is 71% the size of the larger one. Sooo what, you say?

Try the same equation with another resolution - one that seems a little more "uneven" between the length and width. As another example, 1400x1050. In that situation, we'd again take the smaller length (1400), divide it by the wider (1920), and we would be given the result of 0.72. Again, so what? Well, do the same thing with the height. Smaller (1050) divided by larger (1080) is... 0.97!

This means that the length and height of the new resolution, 1400x1050, is not equally smaller to 1920x1080, our resolution. 1400 is 28% smaller, and 1050 is a meager 3% smaller. This is exactly what results in black bars. It's compensating for the amount of unequal space between your computer's settings/monitor, and the settings of the window within your monitor, so to speak.

So, just use that basic math trick to find which lower resolutions are "friendly", or even with yours, and you'll be good... as opposed to logging in and out after trying a new resolution each time to get rid of those pesky black bars. :x