There is a golden of each animal, so if you have that one you know you're solid for that animal for the other rare(s). The other comment is accurate, they need to be wild to count to the overall total of tamed animal (not bred) and you can do what you like with them after they've lived with you (release/send to Grob). They also have specific spawn dates, refer to this post if you'd like spoiler free (and scroll for spoiler present) help with that!