OK here’s what I was able to figure out so far.
1. The Pet page is created as a simple, empty page and the Pet Manager plugin relies on it to be named “pet-list” (by default, the title is “Pets”). During my testing I have changed the page name (since we’re using nice permalinks) to match the language of our blog which broke it. Renaming it back to “pet-list” restored the functionality. So the plugin probably needs some better way of binding the “pets” page – either add an option to allow choice of any page, or use a shortcode?
2. The Add a Pet page is created as a page titled “Add a Pet” with no specific name, but contains a shortcode [pet_shortcode] so I guess you could turn any page into “Add a Pet” page by using this shortcode.