What is a name for a bowling session?

'Session' is the best term to describe the time spent Bowling. It can also be called a practice or a practice session. Practice with a purpose is called training, or a training session; if a teacher or tutor is involved, it's a bowling lesson.

A tournament session is known as a squad, and a league session is often just called 'league' (as in, "Do you have league tomorrow?").