The kind of meaning we are talking about
I agree with @user21102's comment that before we even talk about the psychology of meaning, we need to be clear on what we are talking about: the 2 very different kinds of "meaning":
- What a sign communicates to you beyond its physical properties, such as the meaning of a word, gesture, work of art
- What we human beings generate as an interpretation of our life experience, our existence, our relationship with another person (friend, child, spouse, etc.), our relationship with a deity, etc.
In this answer we are talking about #2.
Cognition vs. Metacognition
Then we need to be clear what "metacognition" mean, esp. since you didn't provide a definition. For this answer, I'll use the American Psychological Association's definition: "awareness of one’s own cognitive processes, often involving a conscious attempt to control them."
The answer: cognitive level
I think I'm still representing your question properly by rephrasing it: "Is the generation of meaning (#2) happening on the cognitive or metacognitive level (APA definition)?"
I'm going to answer: cognitive level, reserving the metacognitive level as when we do second order introspection by historically reviewing the mental processes we used and how those work when we generated / discovered a meaning, for example by reviewing whether there is a religion involved, what emotions led to our generating the meaning, what parental ways of thinking we inherit unconsciously, what the culture says (especially advertising!) how we should assign meaning to certain experiences, etc.
I think it is safe to say, regardless of what philosophy / religion one holds, that human beings are meaning-generating animal. Physicist Michio Kaku, in his 5 minute answer to the questions posed by the interviewer in the PBS Closer to Truth Search for Meaning topic series, describes the propensity of humans to generate meaning as resulted from a "gene" which possibly resulted from an evolutionary benefit and how this can be "tested" in laboratory setting. Since it is part and parcel of being human, we can then safely say it's an integral cognitive function of our brain and/or our soul (for those who hold the view that there is an immaterial aspect of us).
So I think this is on the same level as our linguistic ability, our "language" gene so to speak. Once our psyche generates meaning, we then use our language function to express that meaning in words, which may still be a first-order activity (albeit a subsequent step to the generation itself) in contrast to the metacognition level, which is definitely a second-order activity reflecting on the past discovery of meaning. (By the way, this answer is on the metacognitive level).
Where does cultural influence come into the process? I would say it becomes an integral part of our brain/soul by means of the "input" we process with our senses AND the cognitive processes we used to "interpret" our sensory input (what other people say, what we read). The result is a change in the neural pathways of our brain (AND ALSO a change in the memories of our soul for those who believe the existence of an immaterial element). As people grow up, their meanings change, which is consistent with the above theory of how the same cognitive function produces a different & richer "output"; richer because of the accumulation of life experiences, some of which can be "life-changing" / traumatic such as the death of a child, the onset of cancer, the bankruptcy of one's business, the infidelity of a spouse, etc.
