I'm on GoDaddy and using their Installatron and configured for automatic updates, but I've never seen them actually try and push a new version until after it's been out for at least eight or twelve weeks. I've almost always ended up being ready to update manually myself, before GoDaddy / Installatron has ever actually tried to push the next version to me, even though I'm configured to allow them to do that.
But "I'm on a shared server with more people" (implying what you/they update phpBB to might somehow affect everyone else, or that at minimum the Installatron settings are somehow "shared") is not something I'm familiar with on GoDaddy. I thought I was
on GoDaddy's rock-bottom, cheapest shared hosting service, but it sounds like maybe you found one level lower.
Or maybe your GoDaddy support agent was just telling you stories.
(EDIT: Just looked at my GoDaddy Installatron panel in order to show david63 a screen shot, and lo and behold they actually are offering phpBB 3.2.7 already. They hung back for months
on 3.2.5, and for months
on 3.2.4, and every version since 3.2.2. But I agree, they are pushing phpBB 3.2.7 "already". Guess I need to buckle-up and hang on for the ride.)
Annnnny way, I don't have a good explaination for why the "STYLES MANAGEMENT" and "LANGUAGE MANAGEMENT" sections are missing from your ACP Customize tab. But my first and second guess would be that it relates to one of the extensions that is installed, and that one or more of the extensions are unexpectedly affected if not outright incompatible with phpBB 3.2.7. (Regardless of why phpBB 3.2.7 has shown up without asking you first.)
So I would try disabling those extensions you can see on ACP Customize; one at a time to see what the impact to the ACP Customize tab is. If you identify one which seems to be the culprit, research whether there is a phpBB 3.2.7-compatible update for that extension, and go through the extension update process. You should really check that for all of the installed extensions, but start with whichever one seems to be causing issues, if disabling one of them allowed the ACP Customize tab to return to normal.
That style.cfg you're showing at the root of the /styles directory is also not expected. Not that I know this would cause any undefined behavior in phpBB, but maybe go ahead and remove the style.cfg and the Milk_v2.zip that are sitting there, just in case. Also, browse into your /styles/proSilver and /styles/Milk_v2 directories and just make sure that the contents of those styles do in fact appear to still be there. (As opposed to being empty directories now, or other unexpected content.)
If none of that pans out or identifies what's wrong, my own next step would be to download your current board's files over FTP, and then compare what you downloaded against the official phpBB 3.2.7 installation set you download from here on phpBB.com. I've actually done that back when I allowed an Installatron-delivered phpBB version to be installed, and I've never had a difference between "official and Installatron" phpBB files. But maybe their update attempt actually failed part way through or something, and has left you in some "partial 3.2.5 and partial 3.2.7" state.