How to switch tabs with a German keyboard

No, there isn't a way to do this. As an alternative, I would suggest enabling multitasking gestures (Settings > General > Multitasking Gestures). With them on, you can switch to the next/previous app with a four-finger swipe, instead of bringing up the multitasking bar. You can actually swipe through all of the apps in the multitasking bar, in the same order (order of last use). Note that getting to an app through this gesture does not change their order, so if you swipe left to get to an app, you will swipe right to go back.

⌘ Cmd+` works for me with the same versions of Chrome and OS X, so it might be something specific to your installation.

You could try temporarily resetting Chrome's preferences or disabling extensions. Or change the shortcut from System Preferences or test if it works on a different account.