You can think of character customization as happening through three distinct avenues:
1) Texture/Material Change: This is when the model remains exactly the same, but the texture/material applied to it is changed. The most obvious use would be changing skin color or hair color (color in general). In the anime art style, many games use textures for faces/expressions, for example. In high fidelity games, the texture can be something like the complexion/wrinkles/details on a face.
2) Partitioned Models: In this case, the body of the character is split into different parts. Different variants are made of each part and are swapped to create different looking characters. The most basic use of this can be seen in many older RPGs like Skyrim where all male characters have one shared body, but the head and neck are a separate mesh, allowing the game to swap between different head models for different races (elves, orks, humans, etc.).
The skeleton driving the character's animations can be one skeleton, but the meshes deformed by it can be partitioned as much as you like. However, the tricky part about partitions is that, if done incorrectly, gaps can appear where the edges of the different parts meet.
3) Shape-keys/mesh deformation: This is what Daz calls "morphs". It's where the mesh is exactly the same but the vertices are arranged differently, creating a different look. You see this in games that use a slider to modify parts of a character. You can use shape keys to make a body mesh look lithe or muscular through a single slider a la Skyrim. You can also use it to change the shape of eyes, make lips look fuller or thinner, make cheeks look plump or gaunt, etc. You can also use shape keys to give a female character a pregnant belly and enlarge her breasts as well.
Shape keys can also be used to animate characters. Most often, they're used to animate faces or objects that are somewhat amorphous, like a slime or goo.
In terms of ease of use, I would say Textures/materials are easiest to use, then shape keys, then partitioned models. That being said, making 3D characters customizable is an incredible undertaking. It's a lot of work and you need good knowledge of modeling, texturing, materials, and rigging to make it work convincingly.