It's finally working now!
And it probably was working before, but I didn't notice it, because for whatever reason the IP address of the Orange Pi changed when using the ssd. When I tried to ssh on the device, I couldn't get any connection, since I was using the IP to connect. When using the new IP of the device (I was checking the web interface of my router and it showed me an Orange Pi 5 with a new IP), it worked like a charm!
Thanks for your input @royk & @igelkotten!
Just to clarify @igelkotten: I didn't use dd in my last attempt at all. I created one single partition on the ssd and did what @royk suggested from the very beginning: Use armbian-install (which is btw also accessible through armbian-config) and then "4 Boot from MTD Flash - system on SATA, USB or NVMe". After that is completed, it asks you if you want to install/update the bootloader on MTD Flash, which I agreed.
Then I didn't already reboot the device. Instead I took a look at the armbianEnv.txt and /etc/fstab to see if the UUID is correct. Everything looked fine, so I rebooted without the micro sd card and apparently the Orange Pi was working the whole time, but using a different IP. Anyway, it's working now and I'm happily using the ssd!
Thanks again for the help!