First of all, thanks for your replies.
I bought this iMac second-hand so I wanted a fresh start and reinstalled it.
After some testing, see below, I suspect an iCloud account problem, specifically Contacts, rather than an iMac (software) problem.
The problem occurs with the i**@**at.nl account. The other one iCloud account (the j) works fine.
Started with the current situation:
i**@**at.nl as main iCloud account > No contact sync, no drop down menu. Calendar syncs oke
j**@**at.nl as a second iCloud account > contact sync, with drop down menu. Calendar syncs oke
Tested on this iMac:
created a second admin user
j**@**at.nl as main iCloud account > contact sync, with drop down menu. Calendar syncs oke
i**@**at.nl as second iCloud account > No contact sync, no drop down menu. Calendar syncs oke
created a third admin user
i**@**at.nl as main iCloud account > No contact sync, no drop down menu. Calendar syncs oke
Tested on other Apple devices:
On MacBook (Catalina 10.15.7)
i**@**at.nl as a second iCloud account > No contact sync, no drop down menu. Calendar syncs oke
On iPhone8 (iOS 14.0.1))
i**@**at.nl as a second iCloud account > No contact sync, Calendar syncs oke