It's the fourth game in the X series. Originally named after the player's ship in the first game, the X-Shuttle. X being short for experimental.

Regarding "Foundations", from the Release Q&A

Why is the game called X4: Foundations?
Owen: I think some people get confused because of the two different uses of the word foundations. I think some people maybe think foundation like a corporation or a charity, while we're more thinking of the building foundation. Something we build off and what the races in the universe are building off. They're still recovering from all the gates shut down and they're finally getting on their feet.
Bernd: It's funny, how we choose names. It was not not long before the presentation actually that we had a long list of possible names and some people like some, but there was no name that everybody liked. Once that we found this name, everybody seemed to like it. Partially for different reasons. But what I like about the names for X-games is always that they leave some things to interpretation just like the X itself. If the game stands for anything, then it stands for the freedom and that the game can be different things for different people.