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.
I forgot to mention that there's a degree to which a certain common noun can be used to convey the 'class' meaning.
Not all common nouns are created equal in this respect.
Compared to 'woman', 'human' is much more productive in representing the class than 'woman' is.
I am human
for example, is almost as productive as
I am a human
in this Ngram.
One extreme case of representing a class itself with a common noun rarely used to represent the class is the use of 'dancer' in the song titled 'Human' by 'The Killers':
Are we human or are we dancer?
My sign is vital, my hands are cold
And I'm on my knees
Looking for the answer
And here's the comment at genius.com for the line 'are we dancer?':
Flowers admits that “are we dancer” is ungrammatical, although it could be grammatical if we interpret “dancer” to be a species of animal, separate from the human animal.
Despite all the complaints about the line being ungrammatical, I'd simply view it as a case of legitimately, albeit unidiomatically, representing a class itself with a common noun (such as 'dancer') rarely used to represent the class.