Is retaining an unused "Mubarak Housing for Youth" apartment, and falsely claiming ownership of it to avoid returning it to the New Urban Communities Authority, considered taking what is not rightfully mine from a religious perspective?

If the state subsidizes apartments and stipulates their occupancy for low-income individuals, then ownership is not permissible except for those who meet the conditions, based on the hadith: "Muslims are bound by their conditions in what is permissible." Accordingly, your brother's ownership of the apartment is not valid due to his not residing in it, and your purchase of it from him is not permissible. It is obligatory to return it to the authority. However, if the state does not subsidize the apartments and does not impose conditions, then your ownership of it is valid, and you may dispose of it, and it is permissible to lie to protect your right to it.