Anyway, an "engine" is a group of coding that serves for a purpose. Think of the game as little "departments". Every department is a group of code that if you want you can call it "engine". You could call it application or simply code. It's all the same. At the end, an engine is what someone considers it's an engine, but it's common to call "engine" to an ammount of code that serves a specific purpose and does it independently. For example, managing the animation blending could be done by an engine, or the ball physics can be calculated by a physics engine. The final render of every frame is usually done by a render engine.
I've worked with many of very different scales but I've never worked with the "big ones". But the idea behind them is similar. A physics engine, for example, may give you the main functions and properties of the objects of a system (geometries, collision detections, forces). A particles engine may have functions and base objects and libraries of emitters, presets, etc... It depends on the purpose of the engine and the scale at which you want to work. Euphoria, for example, is a very big and powerful animation blending and physics engine all in one. It's awesome for specific cases, but also very expensive both in money and in PROCESSOR performance, so may not be used in a posrts game where you have subtle collisions all around.
When someone says "it needs a new engine" it's the same as saying "it needs a much better code", either from scratch or redoing the current code.
Nowaday games are divided into different "engines". The renderes uses to be an engine itself, and is generally bought. In FPS games, it's common to buy different engines for different "parts" of the game and produce a game in a rush. Some collections of engines are actually called engines as well (UNREAL, for example, which includes a lot of engines to produce yourself a complete game). Buying an engine has it pros and cons, being time the main benefit and customization usually the worse con.
In sports games, they have created their own engines because there aren't commercial ones out there that do what this games need. Or there weren't... I'm sure right now Fifa or PES could use (and may be using) a lot of stuff from 3rd party companies.
PES, for instance, is using new mocap data that should go through an application they surely did buy one or two years ago. Then they had to tweak the app to suit their needs to accomplish their vision. That's why the first iteration (pes 2010) with mocap data had so poor results. They just didn't have the code refined enough to maximize the use of mocap. As we all have seen, pes 2011 addresses a lot this. That's because they've developed a lot more their new engine for using mocap data.
To simplify everything, let's call "engine" to things that work as a "department" themselves. That's why we talk about animation engine, physics engine, AI, etc...
The important thing is to analyze things from a programming perspective (at the end, all needs to be coded in a specific way) so I don't really care about the word "engine" that much. Put it when you want as long as you know what you are refering.
Saying simply "PES needs a new engine", for example, it's a stupid phrase if you don't end it by saying something like "to solve the poor blending of animations".
To me, it's quite clear FIFA has a much better engine, or collection of engines

if you want. The problem is that their vision is so far away from what I expect and want from a football game.
PES, in the other side, has half-baked things here and there, and a strong and solid work of "teamwork/individual AI", based on a bunch of parameters that have a great impact in the gameplay. So, their vision is quite nearer to a simulation of the sport, but the final product lacks a lot of refining, a lot of polishing and definitely could use one new (if there is one currently) engine to cope with all the collisions detection and phyisicality side of the sport. And, because it's a game, no matter if the fundamentals are there, you need the game to be credible all the time and right now the game is not definitely great because of technical shortcomings that DO have a great impact on the gameplay, not only in the stethics.