How do we define temperature?
You have to start with the zeroth law of thermodynamics which is all about bodies in thermal equilibrium.
The strange name is because after the first and second laws of thermodynamics were formulated suddenly somebody realised there was another law of thermodynamics which in some ways was more fundamental than one and two, so rather than change the name of one to two and two to three the law was given the name zero.
In simple terms it can be interpreted as follows.
If two bodies a joined together in thermal contact and no heat flows between them those two bodies are said to be in thermal equilibrium.
The zeroth law states that if body $A$ is in thermal equilibrium with body $C$ and body $B$ is in thermal equilibrium with body $C$ then if bodies $A$ and $B$ where joined together in thermal contact no heat would flow - they are in thermal equilibrium.
A statement of the obvious? Yes, but only after you have heard about the law?
Moving on one needs to find a parameter which will help you decide whether or not two bodies would be in thermal equilibrium is the were joined together in thermal contact.
The chosen parameter is temperature and the device use dto measure the temperature is called a thermometer.
So going back to the zeroth law the thermometer might be body $C$.
Joining the thermometer to body $A$ you might get a reading on the thermometer of 52 (no units yet). If you then join the thermometer to body $B$ and get the same reading of 52 then you know that if joined together $A$ and $B$ would be in thermal equilibrium.
To devise a thermometer you use a thermometric property of a substance; that is a property which changes with temperature and then decide on a scale of temperature so that when you take a temperature reading that reading is meaningful to others and you in the future.
In the video when the statement is made that "when we measure temperature we are measuring dU/dS at equilibrium" a thermometric property is being described which at present happens to be thought to be the best for a particular range of temperatures.
The problem is that although the kelvin scale of temperature is the favoured one at present its realisation in practice over many orders of magnitude of temperature is very difficult. No one thermometric property can be used to measure all temperatures and the problems multiply for very low and for very high temperatures relative to the temperature of the room you are sitting in.