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.