This guide will let you know how animal adoption works in Fire Emblem Engage.
You can unlock it around chapter four or five and you have to do a mission to adopt an animal. After doing the mission, go into a post-battle exploration phase and run around the town to recruit animals. The skirmish and the paralog missions will pop up, so if you can’t get an animal, you can always come back later at some point. You will get three or four animals at each location but you need to wait until a skirmish or paralog mission appears at your location to go back in.
After that, run up to a creature and tap the button to adopt them. Then head back to the Somniel and go to the board. Now every single donation will give you an adoption experience once you reach level two and then you will be able to recruit specific animals.
Once you’ve got your animals, go to your stable put them out onto the stable log and a new item will appear from the animals every time you come back from the missions.