No, you wouldn't. Physical copies can be played inon any console, as long as the cartridge is currently inserted in the console (assuming there are no malfunctions). Digital copies, meanwhile, are tied to your Nintendo Account. No matter which Switch you use, you can download and play games you've purchased as many times as you want.
After you transfer your girlfriend's account, her account (and all her progress) will be removed from the old Switch. If you want to access her games, you'll need to create a new account and link it to your girlfriend's Nintendo Account, before you can make your Switch her "Primary Console" and access her games.
The transfer only affects users and their save data. You will need to redownload any games to the new console before you can play them. Miis are also not transferred, but can be copied separately through the Mii Maker in the System Settings.
If you want to be able to play on either Switch and still keep your progress on both, you'll need an active subscription to Nintendo Switch Online in order to have access to Save Data Cloud Backup. Some games are not compatible to Save Data Cloud Backup, and can only be transferred from one Switch to the next, but not synchronized across multiple consoles. Games that don't support Save Data Cloud Backup include:
- Animal Crossing: New Horizons
- Pokémon Let's Go, Sword/Shield, Brilliant Diamond/Shining Pearl
- Splatoon 2
- and many more
Save data from Animal Crossing: New Horizon can not be transferred using the Nintendo Switch data transfer process. According to Nintendo Support:
Save data that is shared by all users on the system, including Animal Crossing: New Horizons save data, will not be transferred with this process.
- An additional process is available for the Animal Crossing: New Horizons game to transfer your entire island and game save data to a new Nintendo Switch system or to transfer an individual resident to a different island on a new system.
- For more information, see How to Transfer Animal Crossing: New Horizons Save Data.