Using "woman" without article: "I am woman"

woman uncountable noun Collins Dictionary

uncountable noun You can refer to women in general as woman ...the oppression of woman.

As in:

I am woman.

As a lyric it is allowed, and it is grammatically correct. As to your question:

I think that "woman" in the song, is similar to "God" in the phrase "In God we trust." Am I right?

Comparison to deity(ies) is difficult!

Man and men are sometimes used to refer to all human beings, including both males and females. Some people dislike this use, but if an artist were to write a song I am man (a variable noun), it would be grammatical too.

Short Answer

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.

Long Answer

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

[Emphasis added.]

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

[Emphasis added.]

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.

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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.

As all answers already commented; the noun woman can either be preceded by the Indefinite Article a/an or by the Zero article _. I think a more appropriate example would be the root word of woman/man; human. one might say I am human to denote one belongs to that uncountable idea of humanity a type of mammal, similarly, one might say I am woman to denote one belongs to that uncountable idea of womanhood a type of human.

"I am woman", in this sense, means that the singer is acting as a representative/personification of the entire class of women.

Upon seeing the first test explosion of an atomic bomb Robert Oppenheimer quotes Hindu scripture with "Now I am become Death, the destroyer of worlds", meaning (I presume) that he feels a responsibility for the deaths that are likely to follow, and, in a sense feels like the Devil, or whatever force other you associate with death.

The reason we use singular instead of plural is to avoid ambiguity with zero article. "Woman has natural right to exploit man because she ..." "Women have natural right to exploit men because they... " In the second example "they" may refer to both.