Is this usage of "woman" correct?
Yes, it is.
I think that "woman" in the song, is similar to "God" in the phrase "In God we trust." Am I right?
No. It's different.
In English, there's the 'common noun' and then there's the 'proper noun'.
According to Oxford Dictionary, Common noun is:
A noun denoting a class of objects or a concept as opposed to a particular individual. Often contrasted with proper noun
Proper noun is defined in the same dictionary as:
A name used for an individual person, place, or organization, spelled with an initial capital letter, e.g. Jane, London, and Oxfam.
Often contrasted with common noun
So, theoretically, these two are mutually exclusive. That is, if a noun is a common noun, it's not a proper noun, and vice versa.
But the problem is, some nouns such as 'god' can act as both depending on context.
In God we trust. [proper noun]
I believe in only one god. [common noun]
If we had an infinite amount of memory, we could be using only proper nouns. But we don't, so we need to be efficient in naming things by first making up classes of things and then naming only the classes, as opposed to naming individual things. Hence, the default use of common nouns.
But when, in a given context, there's only one thing in a class, we can still use that one thing as a proper noun, as in In God we trust.
Now, turning to 'woman', it is normally used only as a common noun, because it denotes a class of a certain type of people rather than an individual person.
Since the common noun 'woman' represents a class of individuals (i.e., it is countable), you need some kind of determiner such as 'a', 'the', 'some', etc. if you're to refer to an individual person, as in:
I'm a woman
Which means that "I belong to the class of women" or "I'm a member of the class of women".
But in the OP's song, the songwriter didn't want to convey this meaning, but something like "I represent the class of women itself".
Since 'common noun' by definition denotes a "class", you don't need any article before "woman" to convey the latter meaning. Therefore, the correct version here is
I am woman
And this use of woman is to be distinguished from that of God in In God we trust in that the former is still a common noun whereas the latter is a proper noun.
EDIT: I've edited out the rest of the original answer, because whether to treat human in I am human as a noun or an adjective might only complicate the matter and even distract some readers from the call of the question.