Any pokémon you have a strong bond with can be considered your partner, anonymous. There is no real limit to how many that term covers, either. So long as you feel close enough to your pokémon that you would do anything for them, then they’re your partners.
So I wouldn’t worry about it, anonymous, unless your team consists of forty pokémon or more. Then perhaps choose six at a time—if not for anything else, because there are League rules about that.