Which similar properties must objects have to sustain quantum entanglement? 
Quantum entanglement occurs when particles such as photons, electrons,
  molecules as large as buckyballs, and even small diamonds
  interact physically and then become separated; the type of interaction
  is such that each resulting member of a pair is properly described by
  the same quantum mechanical description (state), which is indefinite
  in terms of important factors such as position,momentum, spin,
  polarization, etc. Source: Wikipedia

All objects can get quantumly entangled, but to get a more sustained quantum entanglement with a long dechoherence time, the objects has to have equal properties. Do all properties have to be equal or could some properties be different?
For instance: Form seem to be important for quantum entanglement, but is size important? Could a small diamond be quantum entangled with a larger diamond with the same shape?
 A: Any two interacting systems can become entangled in principle. Entangled simply means the two objects cannot be described by two separate wavefunctions, but must be described by a single wavefunction that encompasses both systems.
The problem is not that your two systems don't become entangled, but that the wavefunction collapses due to decoherence and the entanglement is lost. The rate at which decoherence happens increases rapidly as the system gets bigger (strictly speaking, as the degrees of freedom increase). This is why we can normally only detect entanglement for microscopic objects.
I can't get at the original paper describing the experiment because it's behind a paywall, but some Googling for related papers like this one suggests that the vibrational excitations in diamond are unusually resistant to decoherence for technical reasons that I must admit I didn't really grasp. The decoherence time is around 23$\mu$s, which is eternity as decoherence times for macroscopic objects go!
