As Matthew Najmon said the RAW don't really say. But lets just have fun with this anyway
Lets say you want your armor to look like a winter outfit bang done, but then do you have the right to the properties of said outfit?
Do to the RAW I would say NO you get no bonus properties because your "armor retains all its properties" insulation would be one of said properties.
Ok what about now you want your outfit to look like a special outfit only used by some secrete group. That you may or may not know what said outfit really looks.
ehh... GM's call, but!
In the RAW it says "normal set of clothing" so this is a limitation, but how you define "normal" is up to the GM.
In a game world with multicultural clothing, normal maybe normal in one part of the world and weird or even high class in another part of the world. (Think of the Romans and how they prized silk clothing while silk was common to the most of the East part of the world.)
I would advise that the GM pick a part of the world the the magic armor was made from so you access to that set of clothing, maybe even limit it to one type of clothing (even if it's a nobles outfit, which would be out of the normal outfits but it would be cool to have a king switch between armor and noble outfit at will).
the GM can let that whatever you see you can copy as your outfit. (but it's a little off from the original to keep it normal.)
The key take away is that the outfit that is picked must be "normal" wither or not you can change to many different outfits, or just to one outfit should be made by the GM.