yes, especially in Milan..

..people dress up in the office. The key is presentation in Italia. People want to look like they are on top of the game and highly enriched as a culture, so they tend to be very concerned about how they appear to everyone around them.

In off-work situations, such as going to a Bar/Disco/Shopping it is also typical people dress fancy. I even experienced this in an IKEA on a Sunday afternoon!

You will notice when you are there, people have an obsession with staring at each other. It's just a sensitivity that is unique to that region. It might , at first, make you also vulnerable with your own overall appearance. But if you are the kind of person that doesn't really care about your looks, you will eventually mellow out when you understand how superficial and trivial it all really is.

It's just a regional thing. Most urban italians, even if they participate, really don't agree with it, but just do it to stay within social norm. It shouldn't affect you unless you really are concerned about that sort of thing.

However, as most other cities, it's wiser to stay within the dress code at work to insure no potentially harmful impression of the others.