So today I installed a new Drive-Bay that makes a 5.2" Slot on my computer into a dock for 3.5" SATA drives. The installation worked fine, I enabled Hot-Plug in my UEFI-BIOS and everything looked ok. Until I installed my hard drive into the bay and Windows would not recognize it at all.
I went back into the BIOS, this time with 3.5" drive installed in the bay and saw that the drive is showing up in my BIOS monitor just fine. Just to make sure I re-checked the power connections and finally I plugged the drive directly into the SATA port to make sure it's not the Drive-Bay that's at fault. Still Windows 10 would not recognize the drive.
After looking around on the Internet I found this very interesting forum thread on TomsHardware that suggested running the Windows 10 Memory Diagnostic. Well - no clue why it worked - but it did.
After running the diagnostic on minimal settings with only one pass Windows bootet again and the drive was there and it even shows up in the "Remove Hardware securely" section which I need for Hot-Swapping the drive.