Almost all pet games have an inventory of some sort. In games where the player is expected to collect many pets they usually have their own inventory system, separate from equippable and consumable items. Tycoon games have one of the most minimal kinds of inventories, so let's start there. A pet breeding tycoon game generally has one playing field functioning as a visual inventory of currently owned pets, optionally a secondary playing field with pets being sold in your shop, an inventory category for eggs or frozen animals, optionally a secondary inventory category which is a book recording all pets you have bred so far and which sets you've completed and earned an achievement and rewards for, one inventory category for upgradeable tools, an inventory category for consumable items like medicine, vitamins, food, and insta-grow, and the basic one for money and optionally the other basic ones for XP and/or energy/magic. It may not be immediately obvious that the playing field is an inventory, but there is often a limited number of pet slots, whether those are stalls, nests, pots for plants, or simply a capacity limit. It's also often possible to buy additional space, either within the playing field, as additional copies of the playing field, or in the other inventory categories such as the shop and egg storage. Upgradeable tools are technically "key items" or "plot items" which are generally items the player can't sell and which don't get consumed when used. Sometimes these items have their own XP bars and level up with use.
Equippable gear with combat stats (armor, weapons, accessories) is another standard inventory category. Typically the clothing inventory interface include some sort of paperdoll or wireframe representation of what items are currently equipped on the character, as well as the character's stats as modified by the gear. Some clothing systems have a double clothing inventory; in one set of slots go the stat-bearing gear, but there is a second set of slots where more attractive gear or statless clothing can be placed to cover up the not-necessarily-matched-or-attractive gear.
Consumable items in an RPG-type game can include all the breeding and pet food sort of items from tycoon games, as well as scrolls that teach a skill, armor-repair kits and bandages, various sorts of buffs in pill, potion, or human food form, and gatherables/monster drops which are consumed through crafting. Gatherables include things you pick from plants, dig from the ground, or otherwise find while exploring. Drops include things you can get from tending tame animals (wool, milk, eggs, beeswax, possibly leather and meat if the game allows players to slaughter tame animals) as well as things you can get from fighting monsters and also fishing and similar minigames. The consumable items category can also include crafted items which are used to craft more complex items. (Things like string, fabric, boards, bricks, metal ingots, dye, glue, etc.)
Key items in RPG-type games may include quest-related items in addition to tools and even emotes in online games. Like character customizations, special emotes make nice quest rewards.
Most genres have only one or two inventory locations, though a location may be divided into categories. An inventory location may be an abstract place reachable by a menu, a backpack or similar portable container, a bank which can only be accessed from specific locations within the game and probably has usage fees, or a chest/safe/warehouse built or bought by the player at a specific location within the game (commonly in the player's house). In some systems items can be manipulated within the inventory, either by using a tool on an item, or by using one item on another, or by selecting multiple items and attempting to use a "combine" function. In some cases there may be multiple copies of the same kind of inventory; for example each avatar may have their own backpack or the player may have 3 fish tanks.
So, how is the storage internally structured, and how does the user rearrange it or get things in and out? If a player cannot have duplicates of an item, that's pretty simple: they have it or they don't. Let's call this a binary inventory, because the number of each item is either 0 or 1. Tools and upgrades often follow this paradigm. Also, checklists of achievements, such as areas of the map that have been discovered, types of pet that have been captured or bred (often recorded in a book), and many 1-time quest objectives. These may not be physical objects in an interactive inventory, but the way in which the game keeps track of them is like a read-only inventory. In some cases the player may have an inventory size limited by slots. For example if the inventory has 20 slots the player can carry 20 items, but if they try to pick up a 21st they will either be unable to or will be prompted to discard something.
Another type of inventory would be the stack inventory. In this case, whenever a player gets a new type of item it starts a new stack, then all additional items of the same type are added to that stack, up to a predetermined maximum number the stack can hold. 10, 20, and 100 (or 9 and 99) are common stack sizes. In some systems stack size varies to create the impression that some items are larger or heavier. For example 100 feathers might fit in a stack, while only 10 iron bars do. If a game has this type of information it is stored as part of the data for each specific item, though there is usually a default stack size for a generic item. This is related to the programming concept of inheritance, which... is getting off topic. The player may or may not be allowed to have a second stack of that type of item after the first is full; usually this rule applies to the whole system, not individual item types. As with a binary inventory the player may have a limited number of slots; in this case each slot can only hold one stack of same-type items, regardless of whether it is a stack of 1 or a stack of 100.
Inventory type #3 is the weight inventory. Every item has a weight (or in some cases separate weight and volume numbers). The character's stats determine how much weight and volume they can carry. The two numbers might be affected by different stats, for example the character's strength might determine weight while the backpack they have equipped might directly give a volume. The character can then carry as many of each type of item as they want, but if they try to carry too much weight or too much volume they are overloaded (or pinned) and cannot walk until they discard something. Some games don't allow an overloaded character to pick up more items, others do as long as the character can do it without moving. Some games don't actually allow the character to become overloaded, instead giving the player an error message when they try to pick up an item that will put them over the limit. Those usually still need the functionality of being pinned when overloaded, though, because a character carrying a legal weight may get debuffed so that they are now unable to carry what they were already holding. Some games block the trading function on overloaded characters; I've never quite understood the goal of this rule.
Inventory type #4 is the puzzle inventory. In this case the player's storage is a 2D grid and each item has a 2D shape; the player can carry whatever they can fit into the grid. This is intended to simulate the experience that real backpacks and suitcases can hold more when they are packed efficiently than when you just throw things in.
Finally, there is the rummage inventory. This type of inventory displays each item or stack as a 2D image, and each of these images is on a transparent layer. The player can only access items which don't have other items on top of them. This is intended to simulate the experience of rummaging around in a purse or rucksack and occasionally needing to dig out items from the bottom. Also the visual image of all one's possessions lying in a pile may seem more realistic then having them neatly organized in a grid, especially in a low-tech game world.
Oh, a bit more about the playing field as a visual inventory, especially of pets (or plants). This varies hugely based on the individual game. In some games you have a piece of property, usually with a square grid, on which you place all your buildings, appliances, crops, and/or animals. You're usually not intended to be able to fill up all this space, so the limit on number of animals you are allowed to have on your property may be much lower than the number of grid squares you could hypothetically park animals on. But still, your property functions as an inventory, similar to the puzzle inventory described above. In other games there is no such grid. Fish tanks are a nice example. A fish does not generally stay parked in one square of the tank, it wanders all around and can pass in front of or behind other fish and decorations, through the use of image layers. But the tank does have a limit on the number of fish it can hold. In this case the tank is probably programmed as a binary inventory where the tank holds a set number of binary items, each one being a fish. This is likely because each fish typically has individual data, like age and health, so they couldn't be effectively treated as a stack of a single type of fish.
Time to add to your features list again:
[indent=1]- [For the first inventory, describe its location, the kinds of things it can contain, and starting size]
[indent=2]- [For each inventory, describe how it can be upgraded if this is possible.]