I saw a video of someone using an ice cube in bubbling fat and the fat stuck to the ice cube. How does the cold fat stick to the ice?
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Solid fat won’t stick to solid ice, if they are both solid and cold at the time; they will not interact. Placing a cold substance in a less cool fluid will create a layer of frozen (recently fluid) stuff on the surface of the cold solid, and some of the solid at its surface will melt and mix in with the fluid converting to solid from the other side, so you end up with a mixed zone between the two where the one is intergrown with the other and that is what makes that layer stay. A little further away and the fat, even if solid, will easily separate from the contact zone.
The general point is that even though the oil isn’t all that happy being in contact with water, it doesn’t have a choice. The MUST BE some layer of contact between the two once the one is dropped into the other. That zone of contact is a zone of freezing of the liquid and slight melting of the solid, but things happen fairly fast so chemical segregation is imperfect and the zone is a mixture more than a solution.
Very few pure substances, like water or some specific oil, are totally immiscible, so not a single molecule of the one can dissolve into the other, and vice versa. They maximum amounts (concentrations) are just very low, so on a big scale, we can pretend that there is none (too little to make a big difference from our perspective, even if it is still there). Usually on the order of a few parts per million in fairly strongly incompatible substances. Trace contaminants do happen. Truly pure substances are extremely difficult to make.
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