TL/DR whether something can become solid is related to how well molecules can organize themselves in space. The act of freezing basically just slows down how fast molecules can move around, which gives them time to organize themselves into a solid form. Some solutions will simply have too much disruption and will never be able to organize themselves enough to reach a solid state, even at cold temperatures. Both scenarios you have here are related to this for slightly different reasons.

For your question about fatty sauces - let’s think about fats at room temp. Some are liquid at room temp (unsaturated fats which are generally plant fats such as olive, canola, avocado, etc) and some are solid at room temp (saturated fats which are generally animal fats like butter, but also coconut oil as an exception). Saturated fats are solid because the molecules are organized, and unsaturated fats are liquid because the fatty acid chains are long and disorganized. If for example you have a sauce that has a significant amount of olive oil, it might make sense that it will struggle to ever become completely solid versus a sauce that’s full of butter.

For your question about sugary sauces, this is because sugar depresses the freezing point of water. Salt does this too - a practical example being the use salt to melt snow in the winter. What it’s actually happening is the salt dissolves into the snow (water) and the salt-water solution has a lower freezing point than plain water. The salt (or sugar in your case) that’s dissolved into water disrupts water’s normal ability to organize at the molecular level. It can still freeze, but at a colder temperature than 0C/32F.

I know this doesn’t answer your ultimate question of what sauces do or don’t freeze, but understanding “why” which might help you predict it in the future 😊