Animal rights generally allows adopting rescue animals but opposes breeding animals as pets. Animal rights isn't just about arbitrarily assigning certain rights to animals (I'm not saying you're suggesting this), it's a natural conclusion from believing in animal welfare. When someone is using an animal for profit, that becomes the inherent priority and welfare becomes a secondary concern. If a company can increase profits at the expense of welfare they will, or they will be replaced by ones who do. Words like "welfare", "humane", "responsible", etc., become twisted into marketing buzzwords. Even the cliché small farms still have to compete with the larger companies and do things like surgery without anaesthesia or ship animals long distances without rest or water to slaughter.

The term vegan was coined by a group of people who opposed exploiting animals and was later explicitly defined that way by the same group. As the word has become more common, it's started to become commonly used to refer to a diet, but a more accurate term would be following a plant-based diet.