This is not about getting a bug-free installation (at this stage), but about understanding where the problem comes from. Please do yourself the favour to try an unmodified installation, installed on a completely Oblivion-free PC.
I'd say try to reproduce the problem when playing a new game rather than loading an existing savegame, but I'm not familiar enough with the format of Oblivion savegames to say whether or not they might impact the reproduction of your problem. I also don't know, if and in which way mods impact savegames.
If you can reproduce the problem by playing an existing savegame on a modified installation, this doesn't tell you that Oblivion itself is the culprit, only that something in your existing environment causes the problem, it might be Oblivion, but it might be something else, too.

OK - in this case, I have to give up. Perhaps other people have ideas, or can share their experience with such a problem.
My understanding is that all icon files are treated in the same way by the software. That two icon files are displayed differently (more precisely, the software is able to display one file, but not the other) can only be explained by different file formats. If this is the case, then all people having downloaded and installed the same installation files must see the same problem. Or there is something very different going on, but I have no idea what this could be (in generic terms, some kind of incompatibility making it impossible for your PC to display these icons properly).
For this reason, it would be useful, if other people could post similar problems here. I myself never had such a problem, although I played Oblivion from launch time until today, on several PC's, and installing from CD's and from GOG. I can't speak for Galaxy, as I don't use it. But if your direct download is showing the problem, too, then Galaxy can't be the culprit.
By the way, Galaxy is highly unlikely to fix anything - it's simply another process layer, which complicates things and so may cause problems, but never fix them.

PS: I mentioned the registry, because de-installations don't always clean the registry completely, but leave garbage. This applies, sometimes, to Morrowind, and, people mentioned in the past, to Oblivion, too, occasionally. Re-installing over this garbage may have undesired side effects - but I'm not sure this could cause your problem. Generally, it's not a bad idea to check the registry for such garbage when there are problems with re-installed software, and there is no other explanation.
A word of caution: Registry manipulations may cause problems and make your PC unusable in the worst case. I don't recommend to "play" with the registry, unless people know exactly what they are doing.

Yeah, Oblivion will leave stuff in the registry. I've seen it happen. I had the same issue recently with another game and I had to use ccleaner after I uninstalled the game and removed everything I could find. It did actually work and I was able to play the game.
My guess after looking at your screenshot is that something in your setup or system is messing with the display of that icon. It could be an antivirus or it could be how you system is set up to handle situations where it displays an icon like that.

So I downloaded and installed CCleaner. Used it to clean my registry. Accidentally forgot to uncheck certain boxes and ended up cleaning my browser history, too, but I digress. Point is, it's now even cleaner than need be. While I still had the installation file, I actually needed to add the new Oblivion install onto my firewall exceptions, as my computer correctly thought that the new install was a "new application". I did so, made a new character, and used console commands to add that shirt. Problem still persisted. Oh, Elder Scrolls. The real challenge of these games is getting them to function properly.