Of course Age is about your temporal identity and the intersection of it with social structures of discrimination and oppression. This important fact is not yet well-known, but worry not, soon we'll have a field of study and several journals dedicated to it.
People can accrue all sorts of physical changes at different rates (that still fall within a tight distribution). Some people mentally mature faster than others due to experience, biochemical differences, etc. Some people biologically age faster due to genes, stress, injury, illness, or disease. The volume of UV exposure one receives changes your appearance.
Age happens to closely correlate with so many societal concepts we care about: schooling, maturity and behavioral expectations, dating and marriage, career, having children, getting cancer, dying, etc. Age is universally useful and interesting to us.
> So for you Hindus were wrong
Hindus were also right -- in their local context where the word "gender" is defined that particular way. There is no globally right or wrong way to define the word "gender", because "gender" is just a word whose definition is decided by local common consensus, just like any other word in a language.
Different societies have different words for common greetings with slightly different implications on meaning. None of them are right or wrong because these words are simply containers that can hold whatever we want them to hold. And it's up to common consensus to put stuff into those containers.