The part of the equation you're skipping is that the water on Earth doesn't stay in one place - it's in a constant cycle of evaporation and precipitation.   Water evaporates everywhere, and air currents carry the water vapor to other parts of the planet.   Some of the vapor ends up at the poles, where it gets cold and falls as snow.

If global temps are above a certain level, that snow - plus a little extra - melts every summer.   So over time the ice caps shrink, and more liquid water is added to the global system.   That's the state we're in right now.   To be fair, this has been the state of the planet since the end of the last Ice Age - technically we're still IN the last stages of the last Ice Age.  The concept of "global warming" is really about the great acceleration of the warming process that we're seeing now.

Conversely, in a cooling trend the snow at the poles doesn't all melt in the summer.   So every year the poles trap a bit more of the planet's water and prevent it from returning to the sea.   Also, the colder the planet gets, the farther from the poles the "no-melt" zone gets too, and so the ice sheets grow outward as well as upward.   At the height of the last Ice Age, much of North America and Europe were covered in glaciers.   That adds up to a lot of water that isn't in the ocean anymore.

This is the natural process of the planet, it's ongoing every day.  When the overall temperature trends one direction or other, the world's ice/water ratio gradually shifts as well, over thousands of years.   The polar ice sheets have disappeared and re-formed at least five times in Earth's history.   

But nothing in AI suggests an explanation for the completely unnatural scenario of a frozen Earth with a high sea level.

I suspect it's ignorance on the part of the filmmakers, rather than some grand plan, in the case of both AI and Waterworld.   You're right that semi-flooded Manhattan is sorta plausible if the ice caps melted completely.    But "Waterworld" isn't - even with a total thaw there'd still be a lot of land left.