A metaphor is a rhetorical device in which the author compares two typically unlike things. This comparison is based upon analogies or direct comparisons. Take the following example: "my life is a highway." In this metaphor, the speaker's life is compared to a highway. One's knowledge of a highway is necessary in order to understand that the speaker is referring to the idea that his or her life is filled with ups, downs, twists, and turns.
The phrase "shrieking with fear" is not a metaphor. It does not provide either a straightforward or implied comparison. Instead, it defines how something is shrieking. Shrieking is a verb, and fear is a noun. The phrase is simply stating how one is shrieking (a high-pitched scream).
No a metaphor compares 2 different things. Shrieking with fear is describing the scream.
"Shrieking with fear " Is not a metaphor because a metaphor is a figure of speech when a word or a phrase is applied to an object or to an action. So "Shriek of fear" is not a metaphor because it is not applied to anything it is just describing the fear.
No, "shrieking with fear" is not a metaphor. A metaphor compares two different objects (without using the words "like" or "as" because then that would be a simile). Ignoring the preposition "with", the two other words in the phrase are "shrieking" and "fear". These are not being compared to one another, but instead one is used to describe the other. How did the cowardly child on Halloween shriek? Fearfully - hence, "with fear".
An easy way (I think) to test whether a statement is a metaphor is by trying to turn it into a simile. Take the sentence - Her eyes were stars. Here, the comparison is the brightness (implied) of the said woman's eyes versus the brightness of the stars. You can easily transform this into a simile by plugging in the word "like" - Her eyes were [bright] like stars. Back to your question, "shrieking with fear" does not lend itself to the same manipulation. After all, there is no meaning derived from "shrieking with like fear" or "shrieking with as fear".
A metaphor is a figure of speech in which a word or phrase is applied to an object or an action to which is not literally applicable. A metaphor can also be a thing regarded as representative or symbolic of something else, especially something abstract. "Shrieking with fear" is not a metaphor because it is not comparing the shriek to something, its really just describing the shriek.