Not necessarily - as long as it has been granted the correct Channel access, it will be available on all performances for these Channels once it's created.
However, if you wanted to control exactly which performances you would like it to be available for then you would need to link it via the Performances page of the Event.
If after that you would like to link the Product to another Event, then you would need to link it to those specific performances as well.
NOTE - If you need to report on Products sold per performance then they must be linked to performances. If Products are not linked to performances, you will not be able to report on Products sold per performance.