the equal sign comparison

The single equal sign in the C programming language is called the assignment operator. It has a different meaning than in algebra class where it indicated an equation or equality. The assignment operator tells the microcontroller to evaluate whatever value or expression is on the right side of the equal sign, and store it in the variable to the left of the equal sign. This is the explanation of the = sign. Why does the microprocessor evaluate the assignment byte? c = 1?
The assignment byte, string, word etc... Is it to determine if the "1" will fit and is the right context that it should be before it assigns?

No. The assignment operator returns a value so that you can do this:

int a, b, c, d, e, f;
.
.
.
a = b= c = d = e = f = 14

;

There is no checking that the value fits in the type. There is no checking that the value is logical for the type. In the statement above, the value could have been 1234567.890, and the compiler would have not had a complaint.

in the plagurized copy of the reference it does state that the assignment character is tested.

The assignment operator tells the microcontroller to evaluate whatever value or expression is on the right side of the equal sign, and store it in the variable to the left of the equal sign.