For what it's worth, and this is just my humble opinion: I've seen just about everything from flip flops and denim cutoffs to dress shirts/ties on all ages in the MDRs. There are plenty of young people (and older people!) in dress shorts, but the MDR's can be chilly so my family abandoned shorts for dinner after our first cruise. My boys like to dress up a bit for dinner, so DH, DS14 and DS9 wear khakis/Dockers with a polo on most nights. In our experience, that or a polo with dress shorts is what the majority of men and boys wear on most nights. Other cruisers' mileage may vary. I pack a coordinating pair of khakis, a polo shirt, + a dress shirt/tie for formal night for each of them (plus pirate gear for the boys since they still like to do the pirate thing). DH takes a pair of dress shoes but the boys just wear their everyday sneakers and they're set for all occasions. It doesn't take much room and they combine and recycle as necessary; they only have them on for an hour or two at a time anyway and it'd be easy enough to wash them or have them cleaned if any unfortunate food accidents occurred...
and absolutely nobody is going to notice when they wear the same thing to dinner more than once. Can't speak for most women and girls, but I like to dress up a bit too so I usually wear a casual skirt and flats, or on optional formal nights, a dress. There are always plenty of women/girls in nice shorts and capris.
Really, the dress code is very loose, especially for children. You might be turned away if you showed up in swimwear, but almost anything else goes. I wouldn't recommend clothes that are dirty or especially ragged, but for the most part I've never seen anyone on the extremely casual end of the dress spectrum getting the Hairy Eyeball even though they were in the minority. You're more likely to get negative attention if you're loud/obnoxious or letting your kids run wild. Men in tank tops may be an exception, and I kind of get that... I don't especially want to eat next right to a stranger's gently waving armpit hair either.
Enjoy your cruise!