Sure, it’s his movie theater, so the President can invite anyone he damn well pleases. Not only can he invite Spielberg, but he has, as this photo from the always amazing White House Museum website (which has in-depth information and scores of photos on every room in the White House, including the theater) shows:
Since the First Family obviously can’t go down to the Century or IMAX, the family theater in the East Wing is there for them to watch films. I mean, technically, they could go to a regular theater, but it would be such a logistical nightmare that it wouldn’t be worth leaving the White House for something that last two hours. Plus, Presidents haven’t always had the greatest track record in going out to the theater.
The White House theater is a nice size for small groups and has the most comfortable-looking chairs that you will probably ever find in a theater. One of the perks of the Presidency — besides the theater itself — is the fact that the major film companies provide first-run movies for the President to screen in his theater. They can also watch older movies or sporting events like the Super Bowl in the theater.
The theater has other purposes, too. Presidents often use the theater when preparing for debates, press conferences, and, especially the State of the Union Address.