Yes, but in the case of some games, the save data is shared.

I don’t have a Pokémon game on Switch, so I can’t speak to it directly.

On Animal Crossing, the first player goes through an orientation scene at the airport where they are told they won an island getaway package. They create their character, and then they choose an island shape from four randomly generated choices. Once all that’s done, you fly to the island and get to see the airport colour (this matters to some people), the native fruit (determines what recipes you can make, and some have preferences), and who your starter animal villagers (always a Big Sister personality female and a Jock personality male) will be. Sometimes you don’t like one of those, so you reset. I had to reset seven times to get an island I wanted (blue airport, oranges, and I forget who my starters were now — they have both moved away).

So anyway, the second player does not get the orientation and the island picker. Instead, their orientation has them joining the first player’s island. The first player will always be the “Island Representative” and this is the only player Tom Nook (the racoon “boss” of the island, so to speak) will talk to about upgrades for the island and features you can unlock by completing simple quests. The other player(s) will have access to these once the “island representative” has unlocked them, but if they required learning a recipe, the other players will NOT be given that recipe. They can find it randomly in a message bottle on the beach, shot down from a balloon by a slingshot, or from a villager who is crafting something. Or, they can get it from another player, such as a hacked player running a treasure island where you can get ALL the recipes.

Case in point: My wife’s island. My wife bought the Switch for Animal Crossing, then decided she did not like Animal Crossing (it IS kind of a slow and pointless “chill” game), so she stopped playing. She unlocked a couple things, but I could not unlock anything else. So she agreed to let me delete her island and start a new one.

Case in point 2: My island. I’ve been playing my own island since May. EVERYTHING is unlocked. So she can start a new character, move onto my island, and have access to all the features. There may be a few things she can’t craft because she doesn’t have the recipe, and the game is less likely to spawn a recipe the island rep got for free (it’s considered lower priority).

Sorry for the long reply, just wanted to clarify how it actually works. This is intentional: for “personal” games like Pokémon and Animal Crossing, Nintendo fully expects the second player to buy their own Switch and their own copy of the game. Once you’ve done that, you’re both island reps on your own islands, and you can visit each other over a local connection (no paid Nintendo Switch Online account required!). Now this is where it’s super important to make sure your second island does not have the same native fruit. There are five fruits (orange, apple, cherry, peach, and pear) and you will always have a native fruit, your fruit trees only grow this. You will have a secondary fruit your “mom” will send you and other villagers will gift you, and mystery islands have a low percentage chance of spawning. Once you figure out what those two are, you want to make sure the second island doesn’t use either as its native… and hope its secondary isn’t the same either. Then, you can trade fruits, and now you have access to four. You’ll need to trade with at least two other players to get access to all five fruits. (There are also coconuts but everyone is guaranteed to get those.)

For some reason, no. Not for things like Animal Crossings New Horizons. It’s the same island so the same save so to speak. If my twin started playing the game on their profile, they could do whatever they wanted to my island because it’s the same island.