OK, I solved the problem for me!!! and I really liked my conspiracy theory...
And that is what I did:
First I flashed the newest firmware on my AP (in repeater mode). Since I have B1 version I could upgrade from 2.10 to DAP-1360_fw_Rev-B1_v2.11B23_nml. (Do not know what's the status for A1) The upgrading went a bit oddly and it looked as I bricked my AP when I was following the instructions from D-Link for upgrading. However, when I used the "Quick Start" from the installation CD (that one that had to be done initially to activate the AP) my AP was back and had the new fw on board. The settings I exchanged via WSP with my DIR-685 router (Basically, just sets everywhere the same values for SSID, Decrypting, Password,...). Do be honest I do not know if the upgrade was necessary, but it gave me the most important hint, but I will come to that a bit later.
After the upgrade I checked the network again with inSSIDer and again I found one from my router with a Max Rate of 150 Mps and one from the AP with a Max Rate of 54 Mps, damn

I went back to the AP and set it on "n-Modus only" and than a failure message came (which was not the case with the old fw) that "n-Modus only" is not possible with WEP or TKIP cipher. So I set both, my router and my AP to AES. After I had reconnected my client to the wireless network and checked with inSSIDer I finally had a Max Rate of 144 Mps for my repeater signal. As a consequence on my client the speed went up to 72 Mps and the signal was still strong

I will have the system running for a while and see how it will perform or if I now experience similar connecting issues as richard_f.
Sorry again for the misleading...
P.S. I do not know if it matters, but on my router I have set the wireless channel manually.