Shelter 1 gives you Momma Badger
Shelter 2 gives you Momma Lynx
Paws gives you Baby Lynx
Owning all three games gives you Baby Bear
Shelter 2: Mountains DLC gives a fire emote
Shelter Soundtrack gives a Badger howl emote
Shelter 2 Soundtrack gives a Lynx howl emote
Shelter 2: Mountains EP gives another Lynx howl emote
Paws Soundtrack gives a baby Lynx howl emote
Meadow Soundtrack gives a baby Badger howl emote
Lonesome Fog ebook gives a Fog emote
Fables from the Den ebook gives you a rain cloud emote
Blossom ebook gives you a flower emote and skins for Lynx and Lynx cub
Symbols ebook gives you the Wolf
Art of Shelter ebook gives you a unique skin on all animals and 3d sketch emote
Owning every Shelter related product gives you the Eagle. Meadow DLCs not needed for this.

Update 11/4/19
Trails Comic gives you a new 3D emote and a Wolf skin

Update 6/7/21
Shelter 3 gives you Elephant calf

Obviously you should get all the content that gives you animals.
The skins are neat but you will get so many other skins that you probably won't ever use them.
Most of the emotes probably won't see a lot of use. The most you'll use is probably the howls, as they are pretty fun.
I highly recommend getting the Fire emote. Most people use this when they play a faster character and want people to still be able to follow them. Also it's a great expansion for Shelter 2.

The Eagle is really neat in that it's the only animal that can fly super high and indefinitely. It's really good for just exploring or finding stuff for other players. The problem with this animal is you cannot collect essence. It's up to you whether that sounds like something you'd want. I personally don't use her much.