Based on two independent chapters of the manga by Kudara Naizou.

Is there an anime with people who turn into animals or animals that turn into humans?

Flying Witch Though it’s more “accidental” there is an element of this in Flying Witch.
And a similar theme carries through in later episodes. - Inuyasha Inuyasha is half demon, and he turns into a dog-demon during the day. Also his brother, and many other characters have variations of “people who turn into animals”. - Shakugan No Shana When Margery Daw transforms, she becomes a “beast like” creature.
And there are other variations of this throughout the anime, though not common. - Spice And Wolf Holo The Wise Wolf can transform into a giant wolf whenever she needs to.
Hence the tail and character design. - Bleach Yoruichi can turn into a cat, and back into a human when she pleases. There’s not much else of this in Bleach with other characters. - Miss Kobayashi’s Dragon Maid The main characters and support characters are dragons.
They have both a human-like form and can fully transform into a giant dragon if they want. - Fruits Basket A lot of the characters are taken from the Chinese Zodiac.
Which includes snakes, monkeys, a rat, etc.
All have a human form and an animal form.