On Earth is possible to define the position of something using WGS, and the same system is used for navigation. What is used in space instead? Are different systems used for Earth-orbiting objects (the ISS for example) as opposed to objects beyond LEO (Voyager 1/2) due to practical reasons?
While coordinate systems are used for all sorts of things in space flight, from your position in the solar system down to individual instruments, I'll assume for the purposes of this question that you're talking specifically about coordinate systems used for determining satellite position relative to something.
Also, I'm going to assume you're really interested in reference frames, instead of coordinate systems. For the reference frames I'll talk about, Cartesian coordinates are typically used (Keplerian elements are, of course, another popular alternative).
There are two main types of reference frames: inertial and rotating. Specifically, when they're Earth-centered (origin at the center of the Earth, as they nearly always are, except when dealing with measurements from the Earth's surface), the two main classes are Earth-centered inertial (ECI), and Earth-centered, Earth-fixed (ECEF). ECI frames have axes defined by some inertial reference, such as pointing one of the axes at a specific star or constellation. The Earth rotates in these axes (i.e. the ECI axes do not correspond to any fixed location on Earth's surface). Alternatively, ECEF frames rotate with the Earth.
There are many variations on these two frames, and most of them have to do with how they account for the precession and nutation of the Earth.
This is a much more broad category, but generally they tend to be barycentric frames. You could define a reference frame with the origin at the barycenter, or center of mass, of several bodies (usually two, e.g. Earth-Sun or Earth-Moon).
The International Celestial Reference Frame (ICRF) is valid throughout the solar system, and has its origin at the barycenter of the solar system.
Of course, if you have a mission to, say, Mars, you would define a Mars-centric reference frame to work in.