A game is a piece of multimedia entertainment where the writing has to work as a partner with the gameplay (which is programming underneath) and the art. Well, not all games have stories, and virtual toys may have story but don't have goals. But this section is aimed at games that have a medium or high amount of story.
The word in the section heading that it's most likely readers will be confused by is theme. So let's talk about theme first. Above, we've talked about games which are focused on combat or competition, exploring at a relaxed pace or being super-efficient, solving puzzles or building up a small empire. I've mentioned that games also come in flavors like funny, cute, scary, magical, high-tech, and many others. All of these are theme, though they don't get at the heart of theme. Theme is the statement your game makes about what the (game) world is like and what the player's role is within that world. Who the player should be and what they should do to be declared the winner of (virtual) life. All forms of fiction, including games, are interpreted by the player's or reader's brain as life experience from which they may learn something about the real world and real life. This is why fiction is referred to as "the lie which tells a truth".
The details are all made up, but characters act in ways that express truths of human nature, because they are based on the author's experience of themselves and others. The characters must behave in ways the audience finds psychologically and sociologically plausible, otherwise the characters will feel fake and the audience won't be able to suspend their disbelief and get immersed in that piece of fiction. Similarly, fictional worlds, though they can have magic or gameplay conventions that don't match the real world, are based on the creator's experience of the world. They must behave like something in the real world, though it's common to substitute something simpler for something too complicated, random, or slow to be quick fun. For example growing plants in a game is usually much faster and less prone to random disasters than growing plants in real life. And a difficult skill like picking a lock with several tumblers may be substituted for with a simpler locking mechanism like a sliding block puzzle. Fictional worlds must be internally consistent so that players feel satisfied because they are learning to master an interesting new environment, and don't feel like the game is "cheating" or "AI stupid".
(AI stupid refers to a game being unable to recognize what the player is trying to do or refusing to accept a solution that seems logical and realistic from the player's point of view but the game has not been programmed to understand. For example, say a player must combine a string and a stick in their inventory to make a bow. If using the string on the stick works correctly but using the stick on the string results in an error, that's a classic example of AI stupidity.)
What kinds of themes do games commonly express? A game where the player spends all their time fighting, for example, can't help but promote the idea that the way one wins at life is by being the best warrior. Most of us aren't fighters in real life but the message still comes across that traits which are important to winning fights, like toughness, are important to cultivate in oneself. Also that the problems we encounter in life can be thought of as battles, with enemies we ought to attack and vanquish. An adventure game which has puzzles instead of combat has totally different messages: awareness of one's surroundings, creativity in using tools, and manipulative finesse when dealing with other people are connected with success, while straightforwardly attacking an enemy with a weapon as unsophisticated as a sword seems unlikely to work and unwise. Strategy games are about using one's intelligence and awareness of surroundings to directly and forcefully overwhelm an opponent. A time management game where you have to be efficient and fast to survive can make you reflect that you should be acting more industrious and efficient in your real life, and avoid activities that are inefficient or not obviously productive.
Now, games and stories are not about preaching or brainwashing and don't have a strong effect on most people's beliefs. Usually audiences already have their own beliefs about what kind of role they want to play in what kind of world, and will seek out games that deliver a message they are already familiar with because everyone likes some positive reinforcement, and wants to hear more stories of kinds they already know they enjoy hearing. So as a designer the idea isn't really to say what you think people ought to hear, but instead to analyze the games whose stories you love, and why you love them, and how you can create a game world and cast of characters who will be as much fun for your players. As an added bonus you and your team members will be more motivated on writing bits of story and creating pieces of art to illustrate fun ideas.
So, a simple way to get started brainstorming story ideas for your game is to make a list of game stories you have really enjoyed. Novels, anime, movies, and folktales are all good source material too. Who would you want to be in a game, that your players might enjoy being in your game? What kind of game world would you like to spend time in, that your players might also enjoy spending time in? Since this is a pet game, and you've already decided whether you want one or many pets to be used in a combat or non-combat situation, let's get more specific about that! What should they look like and how would you enjoy interacting with these creatures? No need to limit yourself to story ideas if any gameplay ideas are occurring to you too. Scribble all your ideas down, because it's much easier to work with ideas you can see in front of you than ones floating nebulously around in your head.
If you already have a strong idea of what kind of experience you want your game to be for your players, you can cross out things you like but aren't compatible with this particular project. You can always use them in another project in the future! For example, I think breeding systems are awesome, but a breeding system doesn't fit very well with a game where you want the player to only own and use one or a few pets. So if I wanted to make a game which was about a player bonding with one pet, I'd save my elaborate breeding system ideas for a different game design. Or, if I wanted to make a game about the player as an individual becoming the best warrior ever, I would consider making the player a shapeshifter or animal who fights opponents alone, instead of a human who fights with pet companions. And if there are any other player-controlled animals in the game they could be the civilians who make up the player's home base, the pack/herd/flock the player fights to protect and helps to grow by bringing home the spoils of victory.
I can't talk about every possible case here, or even go into detail about different brainstorming techniques. You can use whatever techniques work for you. I personally like starting with a list of everything that occurs to me, then crossing off irrelevant things, adding new relevant ones, grouping these things into compatible clumps, ordering those clumps in order from ones I like most to ones I like least, then trying to the best or second-best and combine the best parts of the others into it, so that I end up with one big clump of compatible ideas that contains most of my favorites. But some people love mind-mapping/webbing/bubble diagrams, and some people like to analyze 1-3 examples of similar games in detail rather than pull ideas from many sources, and some people prefer to do their brainstorming in a conversation with one or more other people... there are lots of ways to brainstorm that work for different people, or the same person at different times. But the end goal here is to add a description of your game's playable character, world, and thematic appeal to your statement of purpose.
Here's another fill-in-the-blank for you: "The player will control/be [number and type of playable character(s)] who will be [profession] who [game's main activity such as fighting, questing, solving puzzles, or crafting] in the [description of game world] world of [world's name]. [Other important activities] will also allow the player to satisfy their urges to [explore/become wealthy and famous/play mad scientist/help someone/be clever/build something/investigate a mystery/save the world/etc.]." And some examples: "The player will be a witch or wizard who learns the craft of brewing complex potions to create unique pet monsters in the dark fantasy world of Monsturbia. Exploring the world to find ingredients provides a peaceful break between frantic potions-brewing sessions where the player must exhibit the speed and efficiency that will enable him/her to become the world's master monster-brewer, showered with wealth and fame." Or, "The player, represented by an alpha wolf, will control him/herself and up to 7 subordinate wolves at a time in tactical combat, fighting his/her way from the humble beginning of the darkest forest of Wulfmoon to its capital city populated with the other most ambitious wolfpacks of the world. Combat alone can't get the player to the throne; only solving the puzzles in the ancient ruins of Paw's Mark, Broken Fang, and other such locations(dungeons) will allow the alpha to gain the wisdom and spiritual strength necessary to carry out the hero's true task: saving the world of Wulfmoon from itself." Or, "The player will control the team of Wordy the Parrot and Arch the Cat as they bumble their way through a series of zany adventures, not-so-willingly helping the other animal denizens of Abcedaria solve their problems. Wordy and Arch must help each other navigate over obstacles and through mazes, as well as maze-like arguments with their stubborn neighbors. Even if this odd pair succeeds at bringing peace to their village, will they ever be able to stop arguing with each other?"