Until now, there have been problems displaying a Congstar operator logo on the iPhone (including the iPhone 5): the SIM cards supplied simply lacked the relevant information. With the Nano-SIM cards (which also ensure that you don't have to cut your own logo) this is now a thing of the past; as an operator, Congstar is correctly displayed - only the way to activate it is a bit laborious. In the letter that came with the SIM card, a 01805 number for 14 cents from the fixed network is mentioned. There called, one is referred to a free 0800er-Hotline: under 0800 755 1528 you can activate the Congstar-Nano-SIM for free.
It would be nice if Congstar would offer the choice already in the order process whether you want to get a normal SIM, a micro-SIM or a nano-SIM - but it is a bit complicated, because you have to order a micro-SIM first and only then you can switch to a nano-SIM. The hotline also seems to be only activating until 8pm, at least the lady at the hotline said shortly before 8pm that it would be the last activation of the day. I keep my fingers crossed for everyone who wants to activate a Congstar SIM for iPhone 5.
Actually, the problem should have been solved already (and was with my SIM card in the iPhone 5 and iPhone 5S), but now it seems that similar problems are occurring again: more than two years after the replacement and the resulting appearance of the "Congstar" logo, the problem is occurring again. If you have an iPhone 5S with a new Congstar SIM at the 9-cent rate (for LTE in the Telekom network) the operator logo is missing. Since this is not a disadvantage, I will not have the SIM card replaced now - but it is still interesting that the phenomenon can still occur after three years and it seems that no sustainable solution has been found in the meantime.