Procedural generation in terms of ED's "stellar forge" isn't random, however it will most likely use some features of computer "random" number generators to aid with it. The important thing to know is that without some very clever coding and external input sources (sometimes even a hardware random number generator), computers cannot and do not generate truly random numbers. However modern systems will put all manner of variable seeds into the calculations to make the numbers effectively random.
For those that aren't aware, a computer's random number generator takes a "seed" value and from that returns an arbitrary value. When the same seed value is given to the random number generator, the same value is always returned (and the next number to be used is generated from this as well). Therefore a random number generator can be used to generate a sequence of numbers that while "random" are always the same given the same starting seed value.
The following is speculation based on how I'd design Stellar Forge, I haven't looked into the ED files at all:
It's no coincidence that Stellar Forge is broken down into sectors, these will be used to control the generation of stars in each sector. Each sector will be tagged as to how the stars in it a generated - either manually placed fixed stars (often with names) or the type of sector that the stars contain and the density of them. For example, stellar birth sectors will have lots of young stars, sectors further out in the galaxy may contain older stars that are further apart. Even with this the number of sectors in the galaxy will be huge, but not an unmanageable number but sectors could be grouped together to further reduce what needs to be recorded.
Within a sector stars where stars are procedurally generated, they will be generated based on the type of sector, star system density and a fixed (random number generator) seed for the sector. This last bit is important, because if this value changes then the stars in the sector will change as well. Most likely the first thing to generated is the number of star systems in the sector, then they will be placed based on this value and, most likely, the types and densities of star systems in neighbouring sectors. This is where the real procedural generation "magic" starts to kick in, if a sector borders a really dense sector then there should be more star systems on the edge of that sector than on an edge which neighbours an less dense sector - in technical terms this is likely to be visualised as a 3D density curve skewing the generation of stars in a sector to the higher number regions, which is a pretty similar technique to various normal techniques however in 3D rather than 2D (these things are much easier the visualise in your mind in 2D!).
Once a star system is placed then a seed value, likely generated from the same as the placement value for data space saving, is used to generate the number and types of stars in the star system and the bodies that orbit them. Again, this is procedural where if a star is a certain type then it there be certain expectations as to the number, size and distance of planets and the age and type of the system will help to determine the type of the planets and whether or not they have other bodies orbiting them or not.
The position of every body in a star system is generated as a time delta given a generated path for the body, therefore at the arbitrary zero point in time "e.g. 1st January 3000", all the system bodies are at their starting positions and then it's a relatively simple process mathetically to calculate the current position of each body in the star system. This keeps a suitably stable and predictable system in place which won't be reset to a starting point every time you re-enter the star system. This will also be reason why the mathematics of system body motion is simplistic rather than truly Newtonian because otherwise the maths gets far too unpleasant for a computer game and the processing power available.
While star systems themselves are also on the move - often at rather high speed, given the galactic scale FD won't have modelled the motions of the stars themselves as the game won't be around for 20,000 years or so in order to be able to observe and motion.