Are the requirements for an initial type rating the same across types?
Actually, it doesn't matter whether you are a Private pilot or an ATP, the requirements for a type rating are in the Airline Transport Pilot and Aircraft Type Rating Practical Test Standards, which apply regardless of the aircraft type.

In the PTS, it says:


  Practical Test Book Description   
  
  This practical test book contains the Airline Transport Pilot and 
  Aircraft Type Rating Practical Test Standards—Airplane.  The Airline
  Transport Pilot and Aircraft Type Rating Practical Test 
  Standards—Airplane includes Areas of Operation and Tasks for the 
  initial issuance of an airline transport pilot certificate and for the
  addition of category, class, and aircraft type ratings to an airline 
  transport pilot certificate. These Areas of Operation and Tasks also 
  apply for the issuance of an airplane type rating to a private or 
  commercial pilot certificate.


You basically cover everything needed for the ATP.  The differences between the different airplanes come out during the practical test just like they do for any other checkride.  The examiner (who will be typed in the airplane) asks you about the things that are specific to each airplane, but this isn't really a lot different than him asking you questions specific to a Cessna or a Piper during your private pilot checkride.

The PTS specifies general Areas of Operation's instead of saying exactly what needs to be tested for every aircraft type. 

Here they are for a type rating:


  Section 1: Preflight Preparation 
  
  Areas of Operation 
  
  I. Preflight Preparation
  Task A: Equipment Examination
  Task B: Performance and Limitations 
  Task C: Water and Seaplane Characteristics (AMES/ASES)
  Task D: Seaplane Bases, Maritime Rules, and Aids to Marine Navigation (AMES/ASES)  
  
  Section 2: Preflight Procedures, Inflight Maneuvers, and Postflight Procedures
  
  Areas of Operation  
  
  II. Preflight Procedures
  Task A: Preflight Inspection
  Task B: Powerplant Start 
  Task C: TAXIING 
  Task D: Sailing (AMES/ASES)
  Task E: Seaplane Base/Water Landing Site Markings and Lighting (AMES, ASES)
  Task F: Pre-Takeoff Checks  
  
  III. Takeoff and Departure Phase
  Task A: Normal and Crosswind Takeoff
  Task B: Glassy Water Takeoff and Climb (AMES/ASES)
  Task C: Rough Water Takeoff and Climb (AMES/ASES)
  Task D: Confined-Area Takeoff and Climb (AMES/ASES)
  Task E: Instrument Takeoff
  Task F: Powerplant Failure during Takeoff
  Task G: Rejected Takeoff
  Task H: Departure Procedures 
  
  IV. Inflight Maneuvers
  Task A: Steep Turns
  Task B: Approaches to Stalls and Stall Recovery
  Task C: Powerplant Failure—Multiengine Airplane
  Task D: Powerplant Failure—Single–Engine Airplane
  Task E: Specific Flight Characteristics
  Task F: Recovery from Unusual Attitudes  
  
  V. Instrument Procedures
  Task A: Standard Terminal Arrival/Flight Management System Procedures
  Task B: Holding
  Task C: Precision Approaches (PA) 
  Task D: Nonprecision Approaches (NPA)
  Task E: Circling Approach 
  Task F: Missed Approach  
  
  VI. Landings and Approaches to Landings
  Task A: Normal and Crosswind Approaches and Landings
  Task B: Landing from a Precision Approach
  Task C: Approach and Landing with (Simulated) Powerplant Failure—Multiengine Airplane
  Task D: Landing From a Circling Approach
  Task E: Rough Water Approach and Landing (AMES/ASES)
  Task F: Glassy Water Approach And Landing (AMES/ASES) 
  Task G: Confined-Area Approach and Landing (AMES/ASES)
  Task H: Rejected Landing
  Task I: Landing from a No Flap or a Nonstandard Flap Approach  
  
  VII. Normal and Abnormal Procedures
  Task A: Normal and Abnormal Procedures  
  
  VIII. Emergency Procedures
  Task A: Emergency Procedures  
  
  IX. Postflight Procedures
  Task A: After-Landing Procedures
  Task B: Anchoring (AMES/ASES)
  Task C: Docking and Mooring (AMES/ASES)
  Task D: Beaching (AMES/ASES)
  Task E: Ramping (AMES/ASES)
  Task F: Parking and Securing