There is not a lot online on how pets are considered for ties to home country, but the fact that they are never mentioned in the lists with options shows they are not considered any strong tie.
While you feel the dog as a family member, immigration officers do not. Likely out of experience as too many people would not return home to get back to the dog and ...
Further to Traveller's Answer, the page linked above contains this text which states the wording that should appear on an Article 10 Residence Card.
Article 10 residence card
An Article 10 residence card is a document which is issued under EU law (‘the Free Movement Directive’) by EEA Member States to non-EEA family members of EEA nationals who are ...
Information is provided on the gov.uk site https://www.gov.uk/government/publications/entering-the-uk-as-the-holder-of-an-article-10-residence-card/entering-the-uk-as-the-holder-of-an-article-10-residence-card covering:
what it is
what you are allowed to do with it
evidence you need to bring with you to the UK
applying for a Family Permit