Most people only belong to one club, so if that's you, you won't see a switcher at all. You'll just see your club's name and crest in the navigation.
If you do belong to more than one, that same element becomes tappable, opening a list of every club you're part of, each showing its crest, name, and your role there. Roles can genuinely differ from club to club: you might be a Team Admin at one and just a Member at another. Choosing a different club always takes you to that club's overview, since you're switching to a completely different context rather than resuming wherever you left off.