Photon: Object in the scene (without instantiation) that references all players. How?

Essentially, my problem is this - I have an object in the scene that is supposed to have references for all the players present in the scene. How can I do that without instantiating that object? So it would be in the scene from the start.

To be more clear: I want an object “Arena” that has references to all the players. But if I make Arena part of the scene it is not shared - each player has own Arena. How can I make one Arena, that is shared by all players, but having a link to each player’s character?

you question basically answers itself: If you want to have an object in the scene then add it to the scene. This way it gets loaded with the scene itself.
As i am aware that this is hopefully not what you meant and your question is just not accurate i will suggest you look into the use of static classes. These will however not be Monobehaviours and thus do not have Start(), Update()… They also can not use Coroutines themselves.