Ok, you have a few rules you have to meet to bring it aboard.
First, NO ice! Ice can mess up the scanner and they will make you dump it all out before letting you in. Don't worry, ice is available on Deck 9 at the drink station and your room steward will bring as much as you want, up to three times a day!
Second, No hardside coolers(think 'Igloo'), those are what they are not allowing.
Thind, The size of your carry on must fit through the scanner to get into the terminal and on the ship after excursions. Disney uses an airport type scanner. The average size for one to fit is 22 inches long by 14 inches wide by 9 inches thick. It can't be much bigger than that.
And lastly, you can check beverages thru. They must be in a suitable container, I would recommend luggage or soft side cooler, but it has to be something that would contain any spills. Avoid using the grocery store bag

Again, no ice and also here more importantly, no glass.
I would also add about the water. All shipboard water from the fountains and taps in the cabins is the equal of most bottled waters. So is the ice that the room steward will provide to you in room up to 3 times a day, if need be. Just bring a sport bottle(s) that you can fill with ice and water from the tap and you're set.
Water on the ship is all generated onboard from seawater thru a process called reverse osmosis which is the equal of the 5 gal bottled water that is delivered to homes/buisnesses. They do not use heavy chemical additives and it's not pumped in from land while docked.
(edited-ok, that better?

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