That thing in your passport normally is a temporary solution so you can travel and have proof for the permit even though they are still working on creating that card for you. You should have gotten a page on how to fetch your card once it's ready and they should have sent you a letter when it's ready for you to pick up. This can take 6-8 weeks normally.
So what you should do is make an appointment with the Amt that granted you the residence permit (probably your local Ausländerbehörde) and simply ask them. I'm not sure what went wrong, but they will either have your card lying around because you missed to fetch it, or they will issue you a new one. As far as I remember that card was dirt cheap. I think it was either 7.50€ or 17.50€. Worst case you will need new id pictures taken for 9.99€ at the next street corner. (Wikipedia seems to contradict this, quoting the costs of the card at about 100€. I'm pretty sure we only paid pocket change when fetching the card, but maybe we paid the larger part of the sum when we applied for it, with all the paperwork, translations and costs at every corner, I'm not sure.)
Please note that that card is not an official ID and you are required to have your original ID papers (probably passport) with you at all times. The residence permit card is not valid without your passport (I think it's even printed on there).
Now as far as clubs go, they can accept whatever they please. If they want to accept that card without your passport, they can. But German government or the police will not accept your residence permit in place of an ID. They want to see both, permit and passport.
Sources (eAT is the Eletronischer Aufenthaltstitel, the residency permit card):
Kann ich mich mit dem eAT ausweisen?
Für den Nachweis der Identität und des rechtmäßigen Aufenthalts gegenüber Behörden der Verwaltung (zum Beispiel Bürgeramt) oder der Polizei reicht es nicht aus, nur die eAT-Karte vorzuzeigen.
Dafür muss der eAT zusammen mit dem Pass und dem Zusatzblatt vorgelegt werden.
Muss ich auch weiterhin noch zusätzlich meinen Reisepass mitführen?
Ja, der eAT ersetzt nicht die Verpflichtung Ihren Pass mit sich zu führen.
I would love to quote the BAMF directly, but their pages are a 404 mess right now, don't know why.