Are all EGPWS-equipped aircraft using the same underlying system?
Yes and no.  When you say EGPWS that is a trademark of Honeywell, so all EGPWS systems are made by Honeywell.  The fine print is the the requirement TSO-C151c is for a Terrain Awareness and Warning System (TAWS).  

The TSO functionally describes the Honeywell system. (Honeywell actually designed it before the FAA decided it was a necessary addition to the rules.)  Since implementing the standard would most likely require the use of Honeywell proprietary information (patents), Honeywell is legally required to provide the product it to all customers on a non-discriminatory basis. 

As a result, even though Rockwell Collins provides the Integrated Surveillance System (ISS) for the B787, the TAWS component of the ISS is sourced by Collins from Honeywell.

That said, Thales was able to patent a similar system in France about the same time.  There were lawsuits, and in the end they agreed to let each other exist.  ACSS (a joint venture between L-3 Comm and Thales) sells TAWS units in the US that is based on the Thales system, so there is an alternative to Honeywell. Thales does sell their system outside the US. 

Some of the primary Honeywell patents for the EGPWS have recently expired or are about to expire. So there may be some new alternatives soon.  Of course Honeywell has been making 'upgrades' beyond the regulatory requirements which they have been patenting.  These may help them hold their market share.

So no, there is not a single system to meet the requirement.  OTOH, there's not much competition either.