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?").