Is every primitive element of a finite field has to be a generator of a multiplicative group?

Well, the answer depends upon the interpretation of the term $primitive element$. As in the famous book by Lidl and Niederreiter, the answer is yes. But if one defines primitive element to be just an element as in the definition of a separable field extension, the answer is negative. One can see this well explained at,

https://math.stackexchange.com/questions/2958747/is-every-primitive-element-of-a-finite-field-of-characteristic-2-a-generator