Ok, I figured that out - my application was missing Directory.Read.All permissions - this got me on the proper track: https://stackoverflow.com/questions/64679429/ms-graph-api-booking-business-adding-staff-members-with-administrator-role
I just do not know how it was able to read my and my colleagues emailadresses, but not the service account, anyway, all of that is not important, as the setting up the business information does not work. When adding administrator I got an error:
New-MgBookingBusinessStaffMember : An internal server error occurred. The operation failed., You don't have adequate permissions to make changes to mailbox Bookings-UserName@bookings.domain.co.uk. Sid: S-1-5-21-3422892061-1135323251-2670905592-11348201 AccessDetails: MbxAccessAllow,
At line:1 char:1
+ New-MgBookingBusinessStaffMember -BookingBusinessId "Bookings-UserNa ...
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+ CategoryInfo : InvalidOperation: ({ BookingBusine...ngStaffMember }:<>f__AnonymousType15`2) [New-MgBookingBu..._CreateExpanded], RestException`1
+ FullyQualifiedErrorId : ErrorInternalServerError,Microsoft.Graph.PowerShell.Cmdlets.NewMgBookingBusinessStaffMember_CreateExpanded
The weird thing is - the admininistrator acctually finally appears in the list, the problem is - I still cannot set the settings. But listen to this stuff - if I go to GUI, open that particular admin card, and - changing nothing - click "Save" - it all works in MsGraph. No need to reload token, nothing, works instantly, like 1 nanosecond after I click "Save" - I can change the Booking settings using MsGraph. Well.. seems it is a bit not stable so won't use it anyway I guess.