Is the gift, which took place with the aunt's approval and included the registration of most of her property in the name of the agent, and some in the name of his sister and her son, considered a transgression of Islamic law, and is it permissible for his brothers to object to this gift?

Your care and looking after your paternal aunt are good deeds, and it is permissible for you to take a customary wage (اجرة المثل) in exchange for the care. However, it is not lawful for you to take more than that through embarrassment or trickery. Furthermore, it is not permissible to harm the heirs through a gift (hiba) or a bequest (wasiyya).

If your aunt is not competent to manage her money (i.e., not of sound mind or is profligate/foolish), then her gift is Islamically invalid. However, if she is of sound mind and competent, then her disposal of her money through a gift is valid.

In summary: If your aunt is of sound mind and competent, and she gifted you the money willingly and contentedly, without coercion or deception, and without intending to harm the heirs, then the gift is valid. However, if she was not competent, or if the gift was without her consent, or if she intended to harm the heirs, then the gift is unlawful (haram) and invalid, and those funds still belong to her. In this case, you are permitted to take a suitable wage for your service to her and for managing her funds.