Would an aviation engineering background be helpful for a Flight Data Monitoring Analyst?
Skybrary's page on Flight Data Monitoring defines Flight Data Analysis:


  Flight Data Analysis is founded on Operational Flight Data Monitoring (OFDM) which in North America has become known as Flight Operations Quality Assurance (FOQA). It is a process which routinely captures and analyses recorder data in order to improve the safety of flight operations. 
  
  Flight Data Analysis. A process of analysing recorded flight data in order to improve the safety of flight operations. (ICAO Annex 6 - Operation of aircraft) 
  
  Operational Flight Data Monitoring (OFDM) is the pro-active use of recorded flight data from routine operations to improve aviation safety. 


So, this position would be focused on analyzing data from the flight data recorders to make sure the aircraft are being flown in safe and efficient manner.

Engineering (both in aviation and in general) is concerned with design. While an aviation engineering background would probably provide some useful insights, it normally isn't necessary for a position that is not concerned with designing aircraft or aircraft systems.

It would seem that piloting knowledge would be more necessary here than engineering knowledge. In order to analyze whether flight operations are being conducted safely and efficiently, you first need to know what constitutes safe and efficient flight. Technical knowledge of aircraft systems would certainly be needed, but probably more on a pilot level than an engineer level. Additionally, statistical analysis experience will almost certainly be needed.

As an example, here are the qualifications listed for a Flight Data Management Analyst position that I found in a search:


  Essential skills and qualifications:  
  
  
  Experience in statistical analysis and preparation of graphical representation of data  
  Hold accreditation as an aircraft accident/incident investigator or equivalent   
  Hold accreditation as quality auditor or equivalent  
  
  
  Desirable skills and qualifications:  
  
  
  Experience as a pilot   
  Sound aviation technical knowledge in Flying Operations, or equivalent
  Experience in analysis of flight data, using flight data analysis systems (e.g. ADI, Aerobytes, AirFase)   
  Experience in utilisation of flight simulator software to recreate flight events  
  Familiarity with current EASA regulatory legislation  
  Experience in undertaking investigations