Is it considered a sin if I raise my voice and bang on the table while with my mother, and say to her: "Don't make that flimsy excuse," out of anger at her repeatedly using the excuse that my brother is like my sister when I forbid her from speaking to someone?

Raising one's voice at the mother and speaking to her harshly is a form of disobedience (uquq), which is a major sin. The Prophet, peace and blessings be upon him, coupled it with associating partners with Allah. Disobeying the mother is more dangerous because showing kindness to her is more emphasized. The duty is to sincerely repent to Allah and seek forgiveness from the mother. As for a woman speaking with a non-mahram man, it is permissible if there is a need and with lawful Islamic guidelines; otherwise, it is forbidden. Preventing the sister from doing so and admonishing her for it is correct, and the mother should not defend her in such a case. A non-mahram man is not like a brother who is a mahram.