A little update on the issue:
I have discovered, that the URL, which links to the machine details if you go from "Filters - Sessions" to the VDA, is missing the siteid. If I add it at the end of the URL the error is gone and I receive data. The URL to the VDA details if you go from "Filters - Machines" uses the presumably correct URL with the site id.
Example
Working URL (via Filters - Machines):
https://
director/Director/helpDesk/machine/details?locale=en_US&machineId=S-1-5-21-2639188934-1231503823-1836216699-1777&originalFrom=%2Ffilters%2Fmachines%2Fserver-os&machineName=
NAME*&siteid=5b69f08c-c873-4033-9cd9-227a26adaf25*
Not working URL (via Filters - Sessions):
https://
director/Director/helpDesk/machine/details?locale=en_US&machineId=S-1-5-21-2639188934-1231503823-1836216699-1777&originalFrom=%2Ffilters%2Fmachines%2Fdesktop-os&machineName=
NAME
That would also explain why the Eventlog Error says "Parameter received is null or empty siteId."
I will provide those findings to the open case now. This has to be a bug.