Yes. This is one of the reasons why different users should use separate Apple IDs for their devices. There is other data that will be shared as well. As you have learned, iCloud will synch the data between all devices logged into the same Apple ID.
If you still must use the same Apple ID between the members of the family then all of you should stop using Contacts on iCloud (and any other data that you do not want shared) by opening Settings ➡️ Apple ID (Your Name at Top) ➡️ iCloud ➡️ Show All ➡️ Contacts: Set this to OFF on all iPhones sharing the same Apple ID but used by different people. Once that is done that each can do whatever they want with the contacts on their device. The contacts will neither be saved nor synced on iCloud.
The other, and a better, option is to use separate Apple IDs for each member of the family on their respective iPhones and then use 👉 Family Sharing.
Axel F.