You're right, of course. The disc implant must be anchored in place just like a total hip or total knee replacement.
Some of the steps used to hold the implant in place depend on the type of artificial disc replacement (ADR) used. For some, the surgeon chisels out a groove in the bone. A special anchoring keel in the center of the ADR settles into the groove. Bone grows around it to hold it in place.
Other ADRs have a different locking mechanism to hold it in place. Special "teeth" on the outside of each part dig down into the bone to hold it in place. A small amount of motion of the implant is needed to allow normal spinal movement. If the ADR is implanted too tightly, then shear forces may cause a problem. The locking mechanism can break or the implant can come loose.
Researchers are still working to tweak the design of their ADRs. Each new problem (or success) helps them decide how to improve the implant. Early on there were many problems with plates breaking or moving. New designs seem to have taken care of these problems. For now there are very few reports of implant migration. In the short run, it appears they are holding in place just fine. Long-term studies are next.