For one of the nicer restaurants on property, Narcoossee's is somewhat casual. I usually wear jeans and a polo shirt and I have never felt out of place.
As to the whole "No bathing suits" thing, I always wondered why they said that because it would never occur to me to do such a thing. Well we were getting of the elevator from Ca Grill a few months back, and there was a family standing at the podium in their bathing suits and tank tops. I nearly laughed out loud, but it definitely explained why the disclaimer. I've considered Disney my second home for 25 years, but that was definitely a first for me