Is marriage to an already married man in a country that prohibits polygyny permissible by Sharia without documenting the marriage, and does it guarantee the woman's rights? And how can the family be told about this matter?

Polygyny is permissible on condition of justice, and no attention should be paid to laws that prohibit it. Marriage without documentation is Islamically valid, but it may lead to the loss of rights and cause disputes, therefore it is cautioned against. If documentation is not possible and you wish to marry, mediation through family members can be sought to persuade them.