You tell me. Units help describe what you modeled, they can't stop you from modeling nonsense.

I heated up your iron by 4 degrees and now it's 20C+4C=24C. I modeled the physical process of heating and the units provide some additional information about that.

> I heated up your iron by 4 degrees

and what happened when you applied the same amount of heat to the greater mass of Aluminium?

What's the difference between intensive properties (tempreture, pressure, density) and extensive properties, such as mass or volume?

Again, you tell me. Units have no meaning without the context of what you're modeling.

I can model the average temperature of your pieces of metal as (20C+15C)/2=17.5C.

  Units have no meaning without the context of what you're modeling.

I fully agree with that statement ! But some addition of units are widely accepted because their associated model is almost universal on earth, let's say adding mass (kg) of two different objects

Yes, that's why we're having this conversation. It's so common that some people reasonably infer a stronger connection between units and physics.

As a vague intuition, I think of temperature as some sort of "heat density". It's not right, but it helps.

You can move the numerical value by 4 but it doesnt mean you can add Temperatures You're correct though that as long as you have a very specific working model, you can add Temperatures thanks to it. But that model doesnt propagate to the rest of physics

The 4C there is not really a temperature, though. It's a difference in temperature.

I guess that's a limitation of dimensional analysis, since they both have the same units and dimensions.

Exactly, 4C is a quantity with temperature units. Units have no semantic meaning without context. Dimensioned quantities have the semantic meaning that you give them.