If you want to capture the animals you need to activate the Trophy Room and talk to the guy (Matheus Streit) on the first floor who has an Ubirajara with him. He will teach you the skill 'Tame'.
Basically, it's like Pokemon. The animal needs to be Vulnerable, or Asleep, or with Fascination. Then you have a better chance of capturing it.
Not all animals can be captured. The aquatic ones cannot.
Love/Hate Ratio[]
The better your Spirit in the game, the easier it is for the animals to like you. If you follow the hunting route (killing several animals), it will be very difficult for you to tame them, because this skill targets players who follow the love route. Even so, if you persist and don't fight back, just keep playing bait, probably at some moment the animals will fall for it and become fascinated.
There are several alternative endings, depending on your choices. And the narrative adapts itself.
To check how is your spirit status during the game, talk with the survivor Melissa Fasano.