In principle a Mode S radar has no more problem with Mode S aircraft operating duplicated addresses that the old Mode A/Mode C technologies would have with aircraft operating duplicate Mode A codes. Indeed the options for degarbling Mode S are considerably better. The address "uniqueness" can be used to help with reflection processing, but only a very foolish radar designer would not include the potential for a genuine duplicate aircraft into the algorithm.
Problems will only really occur when targets with duplicate addresses are at similar range and azimuth from the radar, where there is the possibility of mis-association of returns.
Multilateration systems, like radar, are able to use their own timing mechanisms to separate out returns from all but the closest targets. The effects of mis-association are usually minor and transient unless the targets are in extended close proximity - for example holding in a stack.
ADS-B has a greater problem with duplicated addresses. This is because it has no directional or timing information on which to avoid mis-association of transmissions. But even here a ground station is able (indeed is required) to be able to handle multiple aircraft with the same 24-bit address provided these are separated by more than 6 nautical miles. However the system must do this based upon inference rather than any genuinely independent measurement.
The biggest problem however is for TCAS. TCAS cannot operate between two aircraft with the same address. Indeed the conflicting aircraft will be invisible to TCAS!
Fortunately, allocation of addresses is fairly well managed and monitored. They are out there, but the occurrences of duplicate addresses in the same airspace at the same time are pretty rare.