If race is a social construct, then is identity also a construct? Is identity the concept of something being itself something that exist in the world or something abstract that doesn't exist in our real world? If identity is not based on something material, then could we say mathematics is also a construct and therefore not real? Is logic even real?
I hate to be the skeptic here, but you have to define the word 'real'. If by 'real' you mean anything that is material, as you have written, then anything that cannot be touched by sense experience (empiricism) is not real. Mathematics and Logic would be, by your definition, not real due to the fact that you can't empirically come across any notion of mathematical inquiry. To ask a number's weight, or it's size seems to be redundant as it yields no substantial answers. As for 'identity', do you mean identity as a person of his or her whole self, or identity as, 'I identify as a man.", albeit a woman?
In short, if one is a materialist, that is to take the position that anything that cannot be reached by sense experience is not real, you are stating that the brain exists while the mind does not.
Race is constructed on lineage, and (over) generalisation about lineage groups.
Buddhists say identity is dependent on causes and conditions, which contradicts our intuition of a permanent self. Another way to say the same thing: the conventional intuitive self is not real. We often define our identity contextually, say as belonging to certain groups, undertaking certain activities, and those remain real ways we can state & communicate things about us. Our intuition of a permanent contextless unchanging self is constructed on such data.
Mathematics is about abstraction, generalisation, especially of quantities, space & time relationships, and multiple units of similar things. It aims to be constructed on all possible and impossible sets of generalisations, and studies the consequences of different sets.