That sounds... less than accurate. Will a sponge float in water? It's less dense...

I'm trying to imagine how the result of a close encounter between Saturn and a Saturn-sized bathtub full of water could be anything other than a single merged sphere with a lot of water content.

A sponge is less dense because it’s full of air holes. But yes, any such encounter would likely tear both “masses” apart and form a new planet.

Right, and Saturn is less dense than water because it is mostly composed of air. It's like a sponge with the spongy lattice removed. It's not a solid. (At least, the vast majority of it by volume is not a solid.) That's why it's called a "gas giant".

I was going to say that it would be a star, but apparently (making an assumption about the mass of the saturn-sized bathtub full of water here) it would still be orders of magnitude from ignition.

An Olympic swimming pool scaled for a Saturn-sized swimmer on the other hand... Ignition!

You’d need on the order of hundred times the mass of Saturn to create a red dwarf, so the swimming pool might actually be enough to kick it into being a Sun-sized or even larger star.

Sounds like the kind of ridiculous lesson one files away when they realize their critical thinking skills have surpassed those of the teacher.