Can I record basketball activity in Apple Fitness on iPhone 16?

Yes, you can. Open the Fitness app, tap the Workout tab (bottom of the screen), scroll down the predefined workouts, tap the "+" icon (after the last predefined workout), scroll down to the alphabetic section and tap Basketball.

You might be right. I do have an Apple Watch. On my iPhone, the list of workouts is in a white font, not a grey font like you show:

I couldn’t find any confirmation, though. What are the workout types that you can start?