Can I reuse the system assets and build a character generator of my own?

By all means. The more the merrier. We even encourage it.

It is perfectly fine to take the base mesh, the targets and the system assets pack and use these to create a full character generator or whatever similar functionality you want. All core assets (the base mesh, targets, skins…) are shared under CC0. This means you in practice have no limits on what you can do with the assets.

There are already several platform-specific endeavors of this kind, for example a character generator for Godot.