How to upgrade a phone with only 8Gb. internal memory
Well, I finally found out about what happened, after learning a bit more about how Android works. It turns out that, although the bloatware apps are installed in /system, their updates are installed in /data. In other words: Google and Sony's bloatware not only eat up space in your system partition, but also in the space that is supposedly for the user.

What I did then was to disable every unnecessary app, and hit the "revert to factory version" option (don't remember the exact wording). This way, although you can't delete them from /system nor shrink said partition, at least you free up internal memory. After doing so, we had more than 900 Mb. of free space, and so we could update the apps that my relative actually wants to use.