Is it considered a sin or an act of polytheism (shirk) to be compelled to attach an ID card or sign official documents that include names such as "Abd al-Amir" or "Abd al-Hussein" for the questioner's mother or grandfather, knowing that it is obligatory to name with the beautiful names of Allah?

It is not permissible to be named "Abd al-Amir" (Slave of the Prince) or similar names that denote servitude to anyone other than Allah. However, there is no sin in using the name and signing with it when there is a need or necessity, for there is no shirk in that, and you are not required to change the name. You may say "Abd Rabb al-Amir" (Slave of the Lord of the Prince) when mentioning him, if doing so does not lead to corruption in transactions.