An ARC card is a form of ID that anyone who is the asylum process, whether first claim or appealing should have been sent by the home office within 3 days of the first screening. It is a useful photo ID as it can be used to get a bank account, GP etc
Please follow the link to see how to chase the Home Office if an ARC card hasn’t been received. You can either call the home office team who did the first screening, phone number on the bail 201 form, or fill out the very simple arc enquiry form within the link.
If the wrong information is on the card, there is no option online to challenge it, so may be helpful to highlight the right to rectification under Art 16 GDPR
Perhaps consider a complaint to the Information Commissioners Office? They do have teeth and can compel Home Office to amend info if need be